Is FORTRAN Still Useful Today?

Which flavor FORTRAN would you use were you to write it in that language?

A "modern language" does not necessary exclude FORTRAN. FORTRAN has greatly evolved over the years since I first learned it. And for this particular application, why not use FORTRAN?

Our department is a heavy user of various FORTRAN dialects. For instance, a good friend one floor up models the spectra of stellar atmospheres using FORTRAN on a Mac Mini. Each 6-D layer in one of his arrays occupies over 1 GB, and yet he has no problem working on data sets this size, and with amazing speed - after writing his array memory manager in..um..FORTRAN?

-e

Thank you so much for your response, Indeed I am well aware that Fortran comes like ice creams in all varieties. Mine is probably 'Pistachio-nut' A Japanese Variety. I learnt it working for the Fuji Bank back in the late 1970's. It seemed very logical, The CEO Gerald Enderby-Smith had tasked me to calculate global oil production and consumption into the 21st century. blah blah. He came back to me with an astonishing statement, that I found difficult to believe at the time "You know Alastair" he said, "This will mean the average price of a house in the UK will exceed £100,000 before the century is out" as it happened my humble house purchased for £19K rose in value to about £350K. It was a semi-detached house in a back street at that!.... "Why?" I asked, "Because the banks will need to shift to property to maintain currency stability".... Gerald declined an offer from Mrs. Thatcher to become Governor of the Bank of England, by the way. I suspect Mr. Greenspan consulted with him from time to time.

Hi Alastair,

I don't know if you're a WinDoze, OS X, or Linux user - or a user of something else - but if you were to use Linux you'd have the distinct advantage of having full access to the GNU Compiler Collection, which includes various flavors of FORTRAN (and no, fellow Netizens, I'm not shouting. FORTRAN (FORmula TRANslation) is an ACRONYM (Alphabetic Co-location for Reducing Or Numbing Your Memory)).

-e

Hi europium, you will probably guess that I am a masochist, when I tell you I am on Win/LoseDoze XP Home Edition, The Pro version was beyond my threshold of pain. My supplier offered both, at no extra charge, but warned me that most XP Pro Users were unauthorised users, Folk just might think I was just another software pirate, unless I was working from a legitimate business address. XP Home is best for Home use, Pro for Office. he said.

The above doodle, is an attempt to illustrate graphically how 'Trivial' extracting the factors from a 'PQ' dual composite are, once the 'iterations' of the 'spiral' of pseudo-squares have arrived at a near perfect square. The first 'EUREKA' moment is when the square root is either of the form:- .....5474496.999999747283888..... or alternatively with one 'odd' number side and one 'even' number side:- ...6544746.499999964768..

As a 'thought experriment' try thinking of a perfectly square flat universe, way beyond reality, and starting in the middle, attempting to tile it with little rectangular tiles, each tile's dimensions you are not able to determine, but you know the precise 'area'. Naturally the floor of this gigantic flat universe has conveniently been laid out as graph paper.

There is a stochastic element to the 'squareness test' as one Merrily goes quite dizzy going round this near endless tile laying regime. from time to time a few false positive results will turn up. Those will not be 'in step' with the asymptotic series and can be discounted.

As soon as the tiny square left in the middle is less than 'two sides' of the large square. Bingo.. you send the result into Dear Professor Burton Kaliski and he sends you a cheque. Sounds Easy.....Think on! There are some interesting similarities with 'The Devil's Invention' here, Asymptotes, Super & Hyper, We need to get a grip of those, if we are to understand and forecast global warming, Ocean currents etc. It could be life and death for our species. we might survive, but the ecosystem might not?

Sir Carnegie writes: "...way beyond reality"

-----

Sir C, you have no monopoly on un-reality here. I guarantee it. One shrewd (and deliberately unnamed) fellow on this forum suggests the Universe is possibly surrounded (on the "outside") by a hollow iron sphere. Okay...

"Waiter! Two more hits of acid, please? And make it snappy!"

Your analogy is totally cool, Alastair.

Do check out Fedora and GCC. You can always configure your system for dual-boot, if desired. I'll walk you through it if you like.

Firstly, you'll need to re-partition your hard drive (not to worry, it's not as tricky as you might think...).

But Zerothly, you should do a complete backup of your hard drive (or, at least, a backup of the stuff you really can't afford to lose). You do have a way of writing DVDs on your system, yes? Or CDs? Please say "yes."

Cheers!
-e

It might be wise to just start with a new/second hand computer. My old ADC Allan, bequeathed me a fairly decent tower, with loads of arty-farty 'borrowed' software . They are very strict in the RMP! I don't fancy a court martial. even though a semi-retired Chaplain (there is still bloody conflict on). and Allen is the stepson of Cannon John Carpenter of Richmond ....tut tut! I will wipe the hard drive clean as a new whistle, with no regrets.

The Frank Nelson Cole tale has a bit of a sad twist. In 1974 I was about to propose to my girlfriend formally, and was waiting for the post. Nothing came, so I thought the post must have been held up. The lady in question was a daughter of Thames Board Mills Chief Engineer (UniLever) I spoke posh and she spoke slightly 'estuary' my Family in Scotland adored her. but.....blah blah.....she called it off.....So in total shock I just turned to my textbooks, it was the only way to cope. Frank Nelson Cole saved my life. The tragedy of it,... it was not her decision, she wanted to die as well!

Alastair, if you please, I'd like to ask you to refrain from such abbreviations as "ADC" and "RMP," and to please spell them out. This is mostly because of the fact that I'm not privy to the same social/cultural/military influences as you; being, as I am, an American and, worse (or better), a Texan. (I almost substituted Yankee for American, but the term has a different connotation here in Texas. In Texas, everyone north of the Red River is considered a "Yankee." It's not, in general, a complimentary term, although I use it in more or less a humorous way and not in the pejorative sense as do some here in The Lone Star State.)

Apart from that, I'd like you to consider the possibility that your severed engagement might just have been a blessing in disguise. In my case, I was allowed to follow-through with the engagement and subsequently married the woman who is now my ex-wife. Apart from producing three wonderful gifted, strong-willed children (my genes dominate, to be sure), the marriage was, in my opinion, a complete disaster. Who knows? Your fate could have been far worse - particularly if your would-be in-laws were less than enamoured with you. Remember: when you marry, you marry her whole family. And no, all my ex'es do not live in Texas.

-e

'!3' has always been my lucky number, your post has cheered me up no end. Aid De Camp...but 'Camp' and 'Gay' have new definitions these days. oops. Royal Military Police...Territorial Branch = Cannon Fodder. Consolation Prize, is the best boss anybody could hope for. Colonel aforesaid. Merlin Hay, Lord High Constable of Scotland. Our house of Lords Computer Spokesperson.

Just now the New Labour administration is squabbling with their Left Wing old guard ultra-socialist back benchers. Tony Blair may be forced to throw them a 'sop' i.e. Organise the abolition of the Monarchy. first that requires reneging on the 92 Hereditary Peers left and giving them all the heave ho. Merlin and the Lord High Constable of England might be spared temporarily for legal reasons? Our Lovely Queen is allowed only one day in her life to reign as Monarch, her Coronation Day, after that if Parliament asks her to eat her own children, she must either abdicate or comply with "The Will of the People" Her Majesty has said that if her subjects no longer want a monarchy she will willingly retire with grace. and begs that it is accomplished with the least fuss. What the people wish is really up to folk like Rupert Murdoch, the 'People' always want what the brain-washing media tell them they want. Even Rupert Murdoch has to follow orders from his masters.

The function of the Royal Military Police is to ensure that the Military behave according to regulations. Some hotheads might just step out of line. Then the prospect of unleashed Hell looms. Weaponised crop pathogens, for instance may be provided to mutineers and destroy the world's delicate hybrid high performance crops.....(We read that they were rather careless decommissioning the research facilities in for arguments sake Boratland). So three billion people start to feel the pinch of starvation..... This is a worst case scenario.....The global economy grinds to a halt.....Particulates that have made nice small highly reflective ice crystals in the clouds are no longer belching out of factories and cars. This leads to hyper-global warming...the Greenland Glaciers start melting at a much faster rate, cold water plummets to the ocean floor disturbing the trapped methane. zillions of tons of it. That all feeds back..blah blah. This is what happens when the Military step out of line.... much better not to. As Chaplain my function would be to council Royal Military Police Officers who faced a severe challenge. Revolution at times like this are not in humanities interest. Whatever anyone may say about Mr. President Bush, I for one, will always pray that God helps and guides his deliberations with his own council. It's a tough job. The same goes for all world leaders.

Every day I thank the Lord for Engineers, In my view they are dedicated to the safety and wellbeing of all. That is why I am such a groupie. Solutions carefully thought out, I know for a fact, are feasible and desirable. also most importanly, affordable. There is plenty of time to implement the best of those plans. Winston Churchill once said "Your Country Needs You" ...now ..."The World Needs It's Engineers" .....We can do it! we can make Heaven, not wait for it! Who promised that we did not have to work for it?

You're a Chaplain?

-e

For Confirmation, My notice of dismissal, may have got lost in the post. Ask the Clerk of the Closet if "The Balmy One" has been sacked yet! We receive a crown (25pence) from the Crown as an Annual salary. Just sufficient to purchase enough salt for the dinner table. Perfect. The most valuable pay is a beaming smile from one of her Majesty's subjects. I guess you could say I was a Royalist. ahem..born and bred. Science is based on 'evidence' just a few days ago a dear friend, I am Roman Catholic, he is a Methodist rang me with the sad news that his sister in law Caroline, afflicted at just 40 with cancer, was close to the end,and in coma. Her husband is a Methodist Minister serving in Cornwall. Her doctors had called the Family out late at night as a matter of urgent necessity. (She was in terminal decline) to attend Caroline's bedside. Mt friend Andrew, who lives in Kielder, Northumbria just could not make such a long journey in time, but as we often pray together, and wanting to share his sorrow, he rang me with the news. We spent about twenty minutes quietly reflecting upon happy times spent with Caroline, etc. Then finished with our Lord's Prayer.

Next day, Andrew rang me. He sounded cheerful, "You would never believe it Alastair, this morning when Caroline's bed was empty, at first the doctors thought her body was in the 'mortuary'..... no she was taking a bath, and complaining she was very hungry and what could she have for breakfast" This episode Andrew tells me has been written in the special book Methodists keep in their Chapel "Answered Prayers" ....

Europium I don't even know if I reach up to the standard required of a Christian, yet alone a Chaplain! I do want to share my poor understanding of Christian love, Saint Paul summed it up in his Letter to the Corinthians:- he described love as follows: "Love (agape) is patient, love is kind. It does not envy, it does not boast, it is not proud. It is not rude, it is not self-seeking, it is not easily angered, it keeps no record of wrongs. Love does not delight in evil but rejoices with the truth. It always protects, always trusts, always hopes, always perseveres. Love never fails." Ch13 all 13 verses (my Lucky Number) I 'title' myself Chaplain, that would be a more truthful answer europium. I doubt anyone other than the Military would have me. Europium, your kind offer of help is, dare I say, a godsend. A lady Friend Dr. Dalma Gordon has asked me to compile an Antibiotic Prescription Decision Tree for the Hospital Pathology lab near Colchester that she supervises. I need to get back into a computer frame of mind. It currently has to be calculated laboriousley with a quill pen on parchment vellum (joke)

Oops I nearly forgot, 33 years ago on Valentine's day 1974 I fell in love with Mr. Cole. Thank you Sir. for 33 wonderful years.

Frank Nelson Cole published a number of important papers, including The Diurnal Variation of Barometric Pressure (1892). I needed a step ladder, but the view from your shoulders was awesome.

fortran is used in ansys(UPF) and abaqus,I think it can keep it's advantage for a long time.

Ixhx?, Thank you for your contribution. I once heard that "Fortran" was God's favourite program language. Mathematicians are convinced that God speaks to them on familiar terms. I suppose the creator is entitled to change to another preference and will doubtless inform mathematicians first. I was also told that God allocates a specific number of times a Fortran Programmer can use the "GoTo" function, when the limit is reached, they are run over by a Bus/Whatever.

AC writes: "The intention of the program is to test for 'squareness' the product of the small asymptotic ratios i.e. 405 x 481 = 174,805 and 174,805 x 1881988129206079...etc. is more pseudo-square than if multiplied by 16 x19 or 304,..."

-----

Btw, I'm solving a similar problem at work, except that it is more a "kth-root" problem, where k₁ ≈ k₂ ≈ ... ≈ k_N. In your problem, N = 2.

-e

Hi Europium, perhaps a 'Before/After' illustration might provide a clue. I made this image for last year's RSA Conference in Nice. (Without the fantastically generous support of RSA Security, there probably would not be any conferences?) Have a look at the $625,000 worth of challenge prizes left.(I think?) If I can crack one or two? (they are not easy!) the proceeds go to dear Professor Larry Gadekin's research effort at BetaBatt in Houston.

or???

"501" care for a game of 'Arrows/Darts'? what a great way to learn kids basic arithmetic. Now that some poor fellow at CR4 is going to wonder what sort of damage I might inflict upon the Engineering community with crackpot ideas? I begin to wonder if Guru status would be entirely appropriate for a 'Groupie'...... My ambition is to try as much as I can to demystify mathematics. Professor John P. Boyd makes a strong appeal to Engineers to grasp certain intractable concepts. The best illustration, that I can think of, with bags of lovely cash as an incentive is here before you. A little further examination of this 'asymptotic series' of similar ratios will reveal the dreaded 'evil' divergence. That can in fact be rather a boon. because the set of suitable ratios is conveniently very large where the composite of two primes is also very large. "All roads will lead to Rome" so to speak. here may be more options to choose from than there are atoms in our universe. each one leads hyper-asymptotically to the solution.

Here is the real 'Guru' :-

The Devil's Invention: Asymptotic, Super-asymptotic
and Hyper-asymptotic Series¤
John P. Boyd
University of Michigan
Abstract. Singular perturbation methods, such as the method of multiple scales
and the method of matched asymptotic expansions, give series in a small parameter
² which are asymptotic but (usually) divergent. In this survey, we use a plethora of
examples to illustrate the cause of the divergence, and explain how this knowledge
can be exploited to generate a "hyper-asymptotic" approximation. This adds a second
asymptotic expansion, with different scaling assumptions about the size of various
terms in the problem, to achieve a minimum error much smaller than the best
possible with the original asymptotic series. (This rescale-and-add process can be
repeated further.) Weakly nonlocal solitary waves are used as an illustration.
Key words: Perturbation methods, asymptotic, hyper-asymptotic, exponential smallness
AMS: 34E05, 40G99, 41A60, 65B10
\Divergent series are the invention of the devil, and it is shameful
to base on them any demonstration whatsoever."
| Niels Hendrik Abel, 1828

In this review, we shall concentrate on teaching by examples. To
make the arguments accessible to a wide readership, we shall omit
proofs. Instead, we will discuss the key ideas using the same tools of
elementary calculus which are sufficient to derive divergent series.
In the next section, we begin with a brief catalogue of physics,
chemistry and engineering problems where key parts of the answer lie
\beyond all orders" in the standard asymptotic expansion because these
features are exponentially small in 1=² where ² << 1 is the perturbation
parameter. The emerging field of exponential asymptotics is not
a branch of pure mathematics in pursuit of beauty (though some of the
ideas are aesthetically charming) but a matter of bloody and unyielding
engineering necessity.

"....In later sections, we illustrate hyper-asymptotic perturbation theory, which allows us to partially overcome the evils of divergence,......"

If any CR4 Engineer, strapped for cash, wants to give an engagement ring to a sweetheart, try an old African Trick. Cicada beetles look like fire opals, there are two varieties, one sleeps for 13 years, another for 17 years. that way, they seldom cross-breed. 'prime number magic' if you use your ears to see where they are, and then find one, just place your finger near. The little critter might just climb onto your finger, if he/she likes you, and even Solomon would never find a ring to match. If Sheba turned up and wore the right perfume i.e. 'Chesert Baa' for instance, the little Cicada might just prefer her finger instead. You will be engaged. If you have seen the Film "Born Free" you will have seen Ngobit Estate, Naro Moru, Kenya. now named Eserian Farm (Peace in Masai) The little Totos (children) taught me that trick. It is always open house at Eserian Farm. and we have a lovely tree house looking over a natural salt lick if any honeymooners would like to snuggle up in a sleeping bag after watching game come for a drink. Just ask John or Lillemore Carnegie Tel:- 00351 245 205 209 (Portugal) and a warm welcome will await. Happy Valentines 2007

The "Three" magnified solutions to the right, are a bit like "Goldilocks" the top one is "Too Hot" and the bottom one is "Too Cold" On this Bitmap the problem set for Goldilocks was to Factor just "493" and if she borrowed a sieve from Mr. Eratosthenes, The three bears Cook, she would probably arrive at the solution a lot quicker.

But that BIG number at the top, is far too big for even Mr. Eratosthenes's largest sieve. The universe in this neck of the woods is just far too small to fit such a sieve, although if you don't mind waiting a few years, it can be done with a smaller sieve, especially if you use a coarse "Elliptical Sieve" first to get rid of the rubbish numbers.

Sadly by that time all three bowls of porridge will not just be stone cold, they will probably have set like stone, or else eaten by the naughty mice.

Miss Muffet the spider has an idea. she was weaving a web over Mr. Pythagoras's four poster bed.....Mmmmm. a sort of "Asymptotic" square spiders web.

This is Miss Muffet, and she says the four anchor threads stretch to the four posts of a very near perfect square bed. That Mr. Pythagoras has his afternoon nap in. The bedspread is chequered and Miss Muffet wants all CR4 readers to know it's zillions of times larger that this tiny universe. Also she helped stitch it very neatly, every patch is a near perfect square. Honest!

Miss Muffet has noticed that her square threads get closer and closer together, then further and further appart as she weaves it, then closer, then father, etc. as the web becomes quite large the spread of these compressions & expansions seems to become larger and larger. the last node to node was more light years than Miss Muffet was able to count. Mr. Pythagoras like counting them for her, he says it's much better than counting sheep, and he gets to sleep much quicker.

Oh yes, Miss Muffet nearly forgot to tell you, Mr. Pythagoras has given Miss Muffet strict instructions as to precisely where she can attach those threads, they must be as near square as she can make them, superasymtotically speaking of course.

Aaaaagh. First attempt to copy a spider.

Fantastic! Spot on! A perfect illustration. Thank you Kris. The simplest of problems, can have highly complex ramifications. Frank Ramsey brother of The Archbishop of Canterbury, was a devout 'Atheist' and posed a superlative 'transcendental' little conundrum. (The late great John Milton Keynes owed much of his economic theories to Frank Ramsay)

"What is the minimum number of guests that need to be invited to a party, so that at least three will be mutual strangers, or at least three will know each other?" (Assuming knowing/not knowing to be symmetric)

Five Guests are not enough! if one guest, Kris, knows only two people, each of whom knows a different one of the folk that Kris does not know, the required 'triplet' quorum is not met. The brute force approach is to list all 32,768 possibilities.

The four guest problem is mind-boggling! answer '18' but proved not by listing all the possibilities. Ron Graham and Paul Erdos provided the proof. Raise the ante to a five-some, and nobody knows. perhaps between 43 and 49? They don't make computers big enough for the brute force approach.

A Fields' Medal awaits whoever gives the proven answer. Athletes do press-ups to keep fit, These sorts of problem keep the grey matter in trim. "The use it or lose it" maxim applies. The pleasure is in travelling the road, the destination may be an anti-climax.

I love the simple that becomes complex , and the complex that becomes simple . The party problem will be mulled over tonight (with the same relish that my wife consumes chocolate ). I've read the '2 people on a bus with same d.o.b. ' problem , and also the 'Monty Hall' problem . Both terrific , especially the indignation of many 'educated people to the latter's answer.

You have a terrific way of seeding interest and encouraging the curious .

It is time to sort out the organized chaos that is my desk. Interestingly ,the human web-weaver I posted has a definite structure in her hands despite it's apparent muddle.

For your lad Kris, this is the Sensational Alex Harvey band. The most amazing track is The Faith Healer. please try to down load it it will full you with joy...another that you may be interested in is Next..but the faith healer is ace have a look Karenxxxxxxxxxx (Karen's Lad 'Sam' has jumbled up the 'QUERTY' to make it hard to post on 'His' Computer) Music is a big healer. And the para-diddles may help understand these crazy 'asymptotes' super & hyper? Let the drum beat Roll. (Alastair) P.S. Have Faith.

Many thanks Karen , It's always good to hear of others . I just popped in here to post my latest rantings . I've been slowed down in my ramblings recently , my Uncle is a real conversation hogger ! I Know the name of the band and will look into it as soon as pos .

Now this photograph was taken by Young "Sam Marshal" Christmas Day 2005 with a brand new Sony DSC-R1, I had asked him to work out how all the gadgets & gizmos operated. Sam is a genius at it..... This is not a double exposure, and there are several pics in this series where the Christmas Fairy dissolves into a blur. Now I know 'Fractal Compression' is used to cram more data onto the memory stick..... but whatever the pretty little girl at the left is doing hiding her face, she sure has a nice pair of legs.... "Wishful Seeing" is the Scientific explanation. Mmmmm .....I wonder? I have seen some amazing images formed from the Mandelbrot Set etc. Fractal Magic perhaps. This is NOT an April Fools spoof.

Paul Erdos said that he thought Mathematics probably got more interesting as the numbers got larger, but only computers could work with them, and it was a pity that one could never see what was going on inside the computer. Fortran Fractal Fairies at work perhaps?

Now that's a challenge - being almost a Cyclops poses certain difficulties . There are strange ways of adapting and covering , but some things such as binocular vision are a mystery I will never truly know. With a lot of squinting I can just about taste it , but only tantalizingly so. I even blagged my way through a paper on stereoscopic imagery once - the truth would have been better but more time consuming unfortunately. There is a great site somewhere with optical illusions - I don't recall its name because it was causing too much visual disturbance for me to view at length.

FORTRAN ! Now you're talking (albeit not my language ). I use Q-BASIC all the time , just for the pure fun of it. You can manipulate damn big numbers with it if you use math short cuts (Ok as I ranted before abcd^efgh mod(ijkl) etc). Simon Singh seems to completely miss the point in one of his books. Shake a few arrays with a little algorhythm and bobs your uncle. This does not require a cray super computer with the size you mention. I will look at the problem you posit in more detail later (Ican only indulge so much time in fun), but if you know FORTRAN and can run it I see no problem . It does not take a seriously big number for 'off the shelf' stuff like Excel to judder.There is a square root approach to your quandary which can simplify delivery the goods , but I'm not sure this is the appropriate forum.As the pq gets bigger the task is harder , but there are some odd flukes. I will return to see with interest.

Changing the language will not alter the essential logic of the task. I learnt to walk before achieving the ability to fall on my face .Still, it makes life fun.

You are a 'Gem' Kris, I have just been looking at an "ANSYS" user manual.....my head hurts already.....I am a simpleton, and just stand back in awe and admiration at folk who can juggle with all these subroutines and stuff. I do notice that a big issue still remains, theft of copyright/whatever, Primary Functions like "Print Output" of necessity have to carry warnings like:- c *** Notice this file contains ANSYS confidential information *** I have personally witnessed a catastrophe (The 4GC, fiasco) unfold as some clever clog thinks he knows how to unpick the lock, so that the script can be borrowed...ahem...nicked. A virtual time-bomb may start ticking, resulting later on with several million pounds worth of brand spanking new top of the range Cray Computer being consigned to the scrap heap, and Lawyers popping round to Posh Car Showrooms in anticipation of extra business. I also wonder if all this high security is getting in the way? slowing the whole process down. The impact of this chaos is nearly always falls on Government projects. I will never forget the day I was hauled out of my house early in the morning by two very amused Police Officers who knew perfectly well I had inadvertently overpaid my Full Years Poll Tax by 26 pence (The council payment office had run out of change) Allowing me to collect my bulging file of correspondence and valid receipts. I was thrown into the 'Clanger' to await the displeasure of the Magisterial Bench. I was sentenced to five days imprisonment, nor was I permitted to show my receipts of correspondence to the court. The Clerk brushed my protests aside. "The Computer had adjudicated me liable to non-payment penalties, 'overpayment' was not a circumstance it was programmed for"...... I was saved at the last moment by my solicitor who had I suspect received calls from the Press, a 'Stop the Front Page' News just in storey. "Man jailed for paying too much Poll Tax " . "Life is stranger than fiction, because fiction has to make sense" (Mark Twain)

I had the amusing experience of asking my local council why they could not provide a wheelie bin for garden waste. Response =my postcode "does not exist" , counter response = "where shall I send my council tax bill".Result =bill + wheelie bin within the week . (That's an expensive bin ). For non -UK read £1000 +. Green ? I was livid.

Back to the point (well mine anyway ! ) Q-BASIC has a cult following on the net, and may amuse to look up. I can run it and dump results into (say) Excel if I choose . Admittedly this is academic , but a lot of fun for those of us who resist being drawn into Bill Gates twilight world of endless upgrading. One day being able to use an abacus may be advantageous. See quote below. ( I am minded to upgrade to Newtons 'ocean of truth' quote , but would hate to lose the existing one). Kris

Challenge Number Prize ($US) Status Submission Date Submitter(s) RSA-576 $10,000 Factored December 3, 2003 J. Franke et al. RSA-640 $20,000 Factored November 2, 2005 F. Bahr et al. RSA-704 $30,000 Not Factored RSA-768 $50,000 Not Factored RSA-896 $75,000 Not Factored RSA-1024 $100,000 Not Factored RSA-1536 $150,000 Not Factored RSA-2048 $200,000 Not Factored

Well Kris, Perhaps the CR4 team can disprove the common assertion that "No Deterministic Method exists to factorise large composite primes" My Fellows in Applied Mathematics have a belief that the "SF" (Supreme Fascist, Paul Erdos's Chum) is NOT amused by such arrant twaddle. We just have to access the correct page number in the SF's Book. and all will be revealed. This year's RSA Security Conference, This time in the City paved with Gold, is to be the largest ever. Whoopee! But let's all think first how we reduce this little wee problemette to it's fundamental basics. Remember each little additional 'bit', doubles the complexity. or put another way reduces the size of the 'needle-hole' that we have to thread. Finding that needle in a proverbial haystack several orders larger than the totality of the cosmos, is a bit tiresome. There are clues, so that makes the search rather fun.

Don't misunderstand me, there is not and never will be a determenistic solution to the pq problem , primes just don't allow it. That is my opinion of the 'my computers bigger than yours' approach . I find it of academic interest (for various reasons). Short cuts yes , easy answers no.

I think , in a practical sense the point is moot. History bears witness to people who thought they had the perfect scheme. A 'pixie' can move with no discernible footprint. As to the ongoing RSA challenge -fools errand I say. Then again the greatest of people fall victim (newton + alchemy etc).

Will check this one again Alastair , Kris.

http://www.designinference.com/documents/2004.01.Irred_Compl_Revisited.pdf

Searching for Ramanujan's ubiquitous "Anti-primes" I stumbled on a highly esoteric and even spiritual blog-spot. My eye caught a familiar reference to the Irreducible Complexity debate. The great Number Theorist GH Hardy is quoted as saying that any damn fool could pose a problem incorporating 'primes' that even the wisest sage could never answer. From my analysis of the "PQ" problem, primes are a complete red herring, they are utterly irrelevant. Their application in cypher is used because a dual prime composite has only two unique factors. so what? .....Big deal!....This problem concerns determining the 'integer' dimensions of a rectangle from it's stated 'integer' area.

This little problemette is merely to locate the intersection point of two curves. to make the problem even easier, we know that one of those curves is the 'square' function. 1,4,9,16,25,36,49,64,81,100,121,144....etc. for small "PQ" composites such as say a twin 32 bit or 64 bit "PQ" a half decent arbitrary precision calculator, and a hot mug of coffee, is all that is needed to factorise the 'rectangle' I will post up a worked example later. "A problem worthy of Attack, proves it's worth by fighting back" And this one is a beauty. the word "BIG" takes on a new meaning.

As you pointed out so well Kris, the 'my computers bigger than yours approach' has severe limitations. A souped up 'Sieve of Eratosthenes' only works if you run a computer lab, and can hitch up all the hardware to run the program whilst your students are out clubbing the night away getting drunk and chatting up skirt.

The haystack and needle-hole analogy was deliberate, as both ends of this long composite integer are significant. The illustration posted was limited by Windoze Paint-shop capabilities to a 17 x 31 rectangle. = 527, 19 x 29 = 551, (24 too big) 19 x 27 = 513 (14 too small) note that 27 is not a prime. In this search, even numbers are as acceptable as odd numbers. 6 x 11 or 66, just happens to be a close cousin of 17 x 31. and 11 x 20 is another......Of interest, is that the close cousins expand as the composite expands. With large "PQ"s we may have more close cousins to choose from than there are atoms in the universe, each and every one of them will yield the factors in less than a millisecond of processing time.

For example, take......Hold on, let's find the pocket calculator,....the composite 9,306,487 for instance, That would take a student a short while to extract the factors with a sieve method. Now tell that student a close cousin is 117, (9 x 13) and the problem is solved. 117 x 9,306,487 x 117 = 1,088,858,979 the square root of that is 32997.86325 so let's round up to 32,998, and square that = 1,088,868,004, subtract our 1,088,858,979 and we get, 9025 = 95 x 95, Bingo! problemette solved. The rest is trivial. (See the illustrations for a clue 9P x 13Q = 1,088,858,979, 13Q - 9P = 190 or 2 x 95....simple junior school algebra)

11 x 16, would work, 20 x 29, & 31 x 45, & 42 x 61, etc. and a few in between, every one an instant winner! All are 'Close Cousins'

Asymptotes. Super-asymptotes and Hyper-asymptotes can, I think, reduce even the $200,000 Challenge into submission. I am quietly confident the hurdles are not insurmountable. If we as Humanity are going to migrate to the 'Stars' as I hope our Grandchildren will wish the same, we need to get a grip with complex problems, as a matter of urgent Engineering necessaity, just as Professor John P. Boyd says.

Here is a handy 'Asymptotic' series of ratios, 7/5, 17/12, 24/17, 41/24, 58/41, 99/70. etc. 7 x 7 = 49, & 5 x 5 = 25, 99^2 = 9801 & 70^2 = 4,900, etc. (Pythagoras on the square applies. This series might just cut out the workload by a lot. Like 'LISP' script, we need to stick to integers.)

You've certainly given a lot to consider on this fascinating subject Alastair. In a practical sense , the pq problem is worthless without considering the time to solve.The 'time-to-solve' can be estimated and is long enough to be useful with current knowledge and hardware.

I do believe in addressing a problem on the basis that understanding is its own reward. The journey to knowledge has many incidental benefits. Well worth it I say.

However , the practicalities of life go on , and one has to weigh up the pro's and con's of investing time in understanding proposition (a) versus proposition (b).

To make a bad joke it comes down to who has a bigger p****. !

I enjoy all the comments/lines you place , but I fear I may retire from this one due to a desire for a quiet life (well , ok , I think you outclass me on math !). Meet you in another line dear chap , Kris .

Your company on this lonely little tread has been very welcome Kris, please don't over-estimate my skill, or underestimate yours. The RSA conundrum is a prime example of asymmetric complexity, to coin a pun. You have hit the nail squarely on the head with your "Time to Solve" observation. 128 bit security is more about verification, than encryption. Your plastic can handshake with your bank, and even the fastest computer could not guess the "Rumpelstiltskin/Whatever" cypher in time to launch an attack. Criminals much prefer to circumvent security, go round the back, and break in that way. load up you RAM with junk-ware, so that poor little innocent computer resorts to a temp file on the Hard Drive. (Virtual RAM) with download speeds now at up to 8 Meg, and only a spare 250 Meg of RAM, in half a minute your hard drive has an infection. I just pull my notebook battery out, whenever I notice a seductive link is acting up. crude but effective. poor little HP notebook. I know the close-down routine, but these rascals have disabled it before, to gain a few extra seconds. Fun & Games for mischievous minds. I suspect the last laugh is on them, as they are providing useful information about weak areas of security. They are testing the system at their own time and expense.

The Internet is pure joy for most folk, it is to me. The spirit of cooperation is tremendous, but being robbed is a big downer. I revert to a spoilt little brat mentality, with a metal age in single digits when things go pear-shaped with my computer. I want it fixed now, and I will scream! and scream! until I'm sick, till it is fixed. Then jump for joy when it is.....The only way I know of saying thank you, is to try my best to help identify potential weaknesses. "The Minimal Criminal" test, one of the most powerful tools in a mathematicians arsenal. Preventive maintenance if you like.

Lastly "Vox Populae Vox Dei" Keep your eyes peeled for any "Reach for the Stars" threads on CR4 Kris, "Hope" in part a sense of "Expectation" can gather momentum and bring forth real progress. When a bright 'Halo' encircles our lovely planet, and children point to the night sky and ask "What is THAT Daddy?" perhaps the father will be able to respond, "I don't know Son, shall we go up there for our Holy-Days and find out?"........ YES PLEASE!......Pax Vebiscum Kris, and Blessings to your Kith & Kin especially your lad. may Grace follow him all his days.

"Retire" does not imply closing my eyes .... (Cue music from Jaws)

Now tell that student a close cousin is 117, (9 x 13) and the problem is solved. 117 x 9,306,487 x 117 = 1,088,858,979 ........OOOOPS! 117 x PQ X 117 does Not =1,088,858,979, 117 x PQ does Kris. Stan Ulam's Joke about ageing mathematicians going senile applies, First they forget their theorems, then forget to 'zip up' and are ready to the old folks home when they forget to zip down!...... Thanks Kris, I need an extra pair of eyes.

Feel free anybody on CR4 to use information posted up to enter the RSA Challenge and possibly earn a few bucks. No one would be more pleased than dear Professor Burton Kaliski and his magnificent research team at RSA Labs. Security is not about "secrecy" all the time. If a lock can be picked, we all want to know, so as to improve that lock.

You are an incorrigible tease Alastair ! I mean that in a complimentary way

Unfortunately (or not ? ) the TLA will never disappear.

Appreciate your blessings , your contacts may be better than mine (my Biog is not complete ! ). Aforementioned son has some terrific qualities (ferocious bests roll over as he chats/strokes them ) - people seem to intuitively like him and and feel happier for talking with him.

As to the original point , I am so easily distracted . Somewhere in this bomb-site I call a house there is a book which references Giblin (your question had set my mind completely off at a tangent . Also a while back somebody gave me a book by a University drop out , " So you think you're a Mathematician " or something like that , and the question reminded me of alternative strategies in thinking .Cue change of vector Kris. I really should shovel up the dust around my computer (no joke - you'll see why elsewhere on this site . ).Anyway as Arnie says - "I'll be back"

Kris, You should introduce your lad to "Benedictine" ....someone I know not too far from this tread....has this georgeous pet Bengal Tiger! and they can be quite ferrosious as well. My Uncle David, I think I mentioned, cared for a Pet Lion named Simba. who was more of a pussy cat than a Lion. A Lioness is the dangerous pet. they need the extra protein.

Snap! I was given exactly the same book as well "So you think you're a Mathematician" Also I live in a house so chock-a=block with books and papers etc. it looks like a skip. That book is 'underneath'.... somewhere....I must read it again.

http://www.americanscientist.org/template/AssetDetail/assetid/28331/page/3;jsessionid=baa9...

A Lucid Interval, 'the perils of precision'

Quote:-

"Why should we have to put up with errors and approximations in computing? Why can't the computer just give the right answer?

Sometimes it can. Calculations done entirely with integers yield exact results as long as the numbers are not too big for the space allotted. Often the allotted space is quite scanty—as little as 16 bits—but this is an artificial constraint; in principle a computer can store the exact value of any integer that will fit in its memory.

Integers have the pleasant property that they form a closed set under addition, subtraction and multiplication; in other words, when you add, subtract or multiply any two integers, you always get another integer. Absent from this list of operations is division, because the quotient of two integers is not always an integer."

Mistakes can even cost lives. During Desert Storm, Patriot Missile Defense systems failed, because of an accumulated error caused by approximating a tenth of a second, for which there is no exact 'binary' equivalent. 1/16 + 1/32 + 1/256 +1/512 +1/4096 ...etc. The small discrepancy went unnoticed. "The Fog of War", yet again. I read that had a LISP type program been deployed, the problem would not have arisen, and had a long extended 'soak' test been used, the problem would have been detected in time.

I am a 'fossil' when it comes to computer software, by eyes glaze over when I look at the vast catalogue of code. Back in the good/bad? old days, we had to use a bit of ingenuity with say a Fortran program, in order to save as much memory space as possible. Sometimes delightful insights were revealed. I once needed to generate the Fibonacci Series, http://search.cpan.org/~tels/Math-Big-1.11/lib/Math/Big.pm#fibonacci() I do not know how it's done here? but I discovered this little Fortran ditty:- Enter 'A', Stop, Loop, 'B' = 'A' + 'B' ,Stop, 'A' = 'B' - 'A' Stop. Print 'A', Loop. The 'Philosophical implications of such a simple routine generating such a profoundly significant series astonished me.

The quest for 'Simplicity' in my view should never be abandoned. I feel deep in my bones that perhaps the most complex problems, may better be solved by a simple approach, than by additional complexity. Super-Asymptotes & Hyper-Asymptotes sound terribly daunting. Dr. John P Boyd is the Guru, I am just the Groupie, but if as he indicates, We can all kiss our Bio-System Goodbye, if we don't get to grips with them, I feel a certain sense of duty to try and understand them. I think this 'PQ' problem to be an ideal area to demonstrate their usefulness. Prizes are just a generous bonus from those lovely folk at RSA Security Inc.

One of my favourite (not mine ) Autistic tales ;

Adult :" What should I do if I cut myself ?"

Child :" Bleed " !

I really should have kept a diary of my sons sayings for publication.

I was musing on 3104² - 573² when I recalled some thoughts I'd had a few years ago , but can I find them ?. Oh well , it's just that something struck me at that time and I don't remember what it was. Such is the result of living in chaos. The joy of later discovery will make me laugh (whatever the mad seedling idea was ).

I agree PQ is more than just an idle pursuit .It can generate other ideas. Plenty of discoveries in life have been incidental.

What I'd like to know is does God play dice ?

Hi Kris, You are not alone, I know of a Lady, who chaired a well known Autistic Society, (JABS) who has done just that, and kept a diary of her own son's 'Sayings' I have his picture pinned to my wall. I peer reviewed some of the data sets from Dr. Wakefield et al....MRC stuff...(the catch-up cohort gave a misleading baseline..blah)

One of those 'Teasers' Kris is the little spider post#18.

We Know from simple Pythagoras, that 2 x 'P.Q' + ('P' -'Q') = the square of the Hippopotimus...ahem...that's what little Miss Muffet calls it. have a look, it's a fact.

Sooooo....If we were to make a square out of two 'PQ's....sruare root that, we would get an integer square root with an integer 'remainder'. If we kept expanding that integer square, (adding our remainder, don't forget) eventually we would come to another 'Perfect Integer Square'. Sounds simple, but let's not waste time doing that little operation integer by integer. we have to hit the 1,4,9,16,25,36,49,64,81,100,121,144,169,196, 225, etc. now we also know that the 'gaps' follow a pattern, 4-1 = 3, 9 - 4 = 5, 16 -9 = 7, 25 - 16 = 9, 36 -25 = 11, etc.

3104^2 - 3103^2 = 6207, so 3105^2 - 3104^2 = 6209, we don't need our calculator. also 6208 x 2 = 6207 +6209, I shall cogitate about the significance of 573^2...Mmmmm?,

Yes, I believe God does play dice. A loaded one! He can throw it in as many parallel universes as he likes. then claim his run of a million double six's was just a fluke.

Ooops! I will be forgetting to "Zip Up" soon! Correction 2 x 'PQ' + ('P' -'Q')^2.

In a hurry I printed off your last 2 posts to look at later. That '^2' thing really had me spinning (even though the correction was printed under ) !. I really should open my eyes more . Still , there were benefits ;

The 'Squaw on the Hippopotamus' can leave the critter with 2 short legs and your original formula holds (not the boring 5:12:13 , the 9800:9801:13860 variety). This possibility should have been obvious to me , but as the saying goes I just don't see the wood for the trees. Amusing because it compounded my initial error (they have eyes but they see not ). Now I'm just going to have to look at the distribution of the a:a+1:hmm creature (versus ooh:b:b+1 variety).

Do I do this on paper or modify a program to do it. It'll have to be both ways . As somebody else re-quotes on cr4 "curiosity has it's own reason for existence". For some reason I don't like using m&n to generate the numbers .

To paraphrase the well known saying , what happens when the obstinate fool meets the seemingly impossible ? The answer (whatever it is) could make a nice epitaph one day .

Hi Kris, You seem to be on the right track, I loved that you spotted the 9800:9801: '13860' illustration, it's a fantastic example....spot on...We know immediately that 13860 is even, and the square of an 'even' number is another 'even' number, and the square of an 'odd' number, 'odd'. so also knowing an 'odd' + 'even' = 'odd', this particular "Squaw" is trying to pull a fast one! The triangle/rectangle is pseudo-Pythagorean.Thanks Kris. not in the least boring well cool! part of that 'asymptotic' series that converges to the irrational root of two! in a very 'rational' way. There is a rule of thumb, employed in that series, that will do the same for other irrational square roots, albeit it gets a bit cumbersome as the numbers get bigger.

Super-Asymptotes have the annoying habit of 'Diverging'. and need a Hyper-Asymptote to whip them back into line. Up here in Lancashire, where I live, folk in the Engineering Brotherhood often greet one another with a phrase like "Hi Kris, what do you know?" The answer could be something like "Rolls Royce are going to go under!" to which a '"How so?" response could be given. "Those new-fangled carbon fibre compressor blades may be light-weight, but if they use them at the front end, the first goose to fly into that engine will turn the bloody blades into shaving brushes!" 1970's Pub Conversation.

In this instance, the answer might be " The online Banking World is in deep shit!"...."How so?" ...."Security? Don't make me laugh!"....."Has there been a big robbery mentioned in the news?"...... "No, but unless something is done about the problem soon, the robbery will be so big, newspapers will be given the old 'D-Notice. they will lose their licence if they print a single word about it. Global Financial meltdown could result!"........."You KNOW this?"........Mmmmmm??????

"The man that started WW3" would not make a "nice" epitaph! On the 26th of September 1983 Colonel Petrov recognised a computer glitch, with just seconds to go, he cancelled the fully automated multi-thousand ICBM meltdown. Andropov had pressed the RED button in response to a False Alarm. I was a Rookie Chaplain at Lakenheath at the time..Phew! what a cold sweat! There was a 'Five Minute Window of Grace' We "Lucked Out" yet again as Robert Strange McNamara would have put it!

Asymmetric Complexity, may be OK to secure petty transactions, but not to secure the safety of Humanity! The knot one man can devise, another can untangle. or even just cut. as Alexander did with the Gordian Knot.

The notion that the Big Prizes that awaits anyone who factors these large composites, is a guarantee that we will have advance knowledge of a breach, in my view, is to put far too much reliance on the seductive attraction of material reward. Even then, a criminal would have the option of employing the knowledge to cash in on a reward of a few thousands, and lose the opportunity of gaining millions or even billions. Fraud is BIG business. Read Professor Loretta Napoleoni. Modern Jihad, "Tracing the Dollars behind the Terror Networks" in 2001 these guys had a $4.5 Trillion Portfolio. Her new book "Terror Incorporated" may have modified that assessment?

If "War is just 'Politics' by other means" then "Politics is also 'War' by other means," as well.....Bankrupt a nation, and their Politicians are forced to be very subservient. Let's not risk it.

Naughty QBasic ! Bad QBasic!! . I really should have done 4059:4060:5741

Oh how I blunder along .The match-stick holding my eye open is a log close up.

I've been busy looking for a program I did ages ago . Gave up and redid it , only to discover the original . Maybe just as well , since I could not understand the poorly commented original . It's purpose , for fun , was to multiply big numbers . I did it in the first inst. because Excel lost digits at a certain point (I think 15 offhand). My original was set up for 20 digit x 20 digits just to see . Worked fine (by comparing with my calculator for front and tail digits + length). I upped the game for fun to 70x70 digits and the logic is fine (well it managed 64x65 digits unless Simon Singh is telling porkie-pies ). When naughty Qbasic gets up , his nappy will be inspected.

I surpass myself with mischief at times. He he.

OOh - what did Kris intend. Must have been more than /2 ?!?

I have no earthly idea why I mention this. 4684659 :4684660:6625109 Something like that anyway. 10th t 10,000,000. All predictable even though the series expands fast I can program to churn them out , but can't express the series in a nice formula. That's annoying is that ! This is more a memo to myself. I can't remember why I was looking at this series of triples.

Hi Alastair , I meant to say earlier , do you have Marcus du Sautoy book 'music of the primes' (I'm sure you do ). It's amazing to a scribbler like me - I have picked it up again to try and comprehend better (might be a long task). Another read I like is 'Riddles in Mathematics' by Eugene P Northrop.I'm minded to rework the Brachiostochone problem and submit it to the Challenge Question ! If I can disguise it , the ensuing thread would be fun. I haven't paused to study the various solutions , but would love to hear a variety of explanations (I'm wandering if there is a really good intuitive solution - a al Simon Singh showing (a+b⁾³ ).

I've only just looked at these links (!). Why did they have to confidently announce factors as verified by RSA labs ? Presumably they mean verifying the factors are genuinely prime (?).

No, Kris,

They had to verify that some wag mathematician, had not just posted the answer up on the internet. and the submission was not just a copy. cut-n-paste job. This little puzzle is best left unanswered in a way, because so many students gain immense benefit from 'having a go'.

Totally agree Alastair , that was why I paid no particular notice . A spare minute to kill got the better of me . The answer is slightly irrelevant.

I continue to diverge . I was going to scan in a copy of my latest notes , but the scanner was refusing to copy an A4 page ! I should take the time to figure it out, but it's just an annoying gadget to me.

I have a pattern for the a:a+1:c type of triplet. Sifting up to the 6th in the sequence , I looked back for a pattern in the n,m building blocks . Sure enough the next set works as 137903:137904:195025. There is a curious discrepancy on the second set (n=2,m=5). m 'ought to be 12. I may have only re-interpreted the wheel here but I enjoy spotting all the patterns , even if I don't know where I'm going .

I will continue to muse on this

Kris

A very dear friend, Dr. Larry Gadekin, who I have mentioned before vis-a-vis BetaBatt, (Exciting Stuff for the Future) told me that 'Pattern' is what it's all about.

Also the heart starts pumping more blood round the circulatory system, as the grey-matter requires more energy. that is good for the whole body. I once read that in energy terms, a Final's Exam, and a Marathon, were about equivalent. Some Volunteer Doctors, did their Final's with breathing apparatus. CO2 exhaled = energy expended.

http://www.mathtools.net/Fortran/Converters/

Hi Alastair , the above is probably gibberish in all senses . Can you believe I've never pasted a link before ! Thought I'd try here since nobody else seems to have a fascination for the juicy problem of factoring a product of 2 primes . I really should look further than Qbasic in my own tinkering , but it still holds the same joy as an abacus. I know there is fun to be had in flipping matrices fast and rattling off simultaneous equations , but simple toys are fun . A simple yo-yo or gyroscope can provoke hours of experiment and debate . Our modern obsession with the latest 'must-have' toy detracts from the most basic of building blocks - a pencil and paper. It's interesting to ponder what would have become of some of the Scientific greats if they'd had too many sweets to choose from (Newton picking up grains of sand instead of pebbles etc while the great Ocean...). Ramble finished ! This particular thread is still mulling in the back of my mind !

Hi Kris, and also Europium, if you are passing through?

(Love the new Avatar Uber-Cool)

If you are seeking 'Inspiration' go here:-

http://en.wikipedia.org/wiki/Norman_Borlaug

If you want to amaze your friends with your new-found mathematical aptitude, I am sure your Local Library or Amazon/Whoever Can get you this book:-

http://funwithfigures.com/?hop=coops52

And.....If you want to have fun with bones? extracting square roots?

http://www.qnet.fi/abehr/Achim/Calculators_Napier_rods2.html

He was my Great great....great grandfather. Mary Queen of Scots was told he had made a pact with the Devil, and that is why she refused to wear the lovely ring Napier gave her. "It was 'enchanted' she was warned. STUFF-N-NONSENCE. Napier and the Carnegies, who paid heed to his council managed to treble the output our Estate's Produce, not by a pact with the horned one, but by scientific management. Neighbours were too lazy to follow the example, and called it 'witchcraft' instead!

I have to confess to being rather lazy myself, part of the reason I posted this thread, I have a mountain of calculations to perform. The Big-Calc I use has just two memories. one has to keep pasting in the values, it's endless cut-n-paste, cut-n-paste work. I believe I have an improved test for "squareness".....as these numbers get larger, so do the problems. A method that works with smallish integers, packs up suddenly.(becomes more effort than gain) boo hoo!

For examination of triangles, Search for 'Lola Luna' (But please don't report me) This is serious mathematics, triangles and Pythagoras are crucial.

Just got a copy of Prime Numbers -A computational Perspective . Looks like a good read ( and will keep me out of mischief for a while ! ) - you may find it useful , if you don't already haver it. One thing that struck me is that the authors (Grandall and Pomerance ) don't seem to place value on creating your own basic math routines ( for multiplying , square roots etc ) , neither did Giblin who I mentioned previously . Spending time on an pq related algorithm that hangs on off-the shelf math software seems a bit (dare I say it ) crass. 'Wow , I'll give up here 'cos Excel cant handle big numbers' etc.

My new acquisition is going to be good fun over the summer , and probably beyond. It's said you can gain a reasonable ability with 1000 hours (?) - well I'd been looking for a topic worth the investment of time , and this may be it . Rather esoteric to many , but that's half the fun . The incidental knowledge accrued is well worth it I think.If anything interesting eventually dawns on me , I shall post it.

This may look a little mysterious to most folk, All it illustrates is "P" x "Q" with two zeros added. i.e. times a hundred. Looking at it, I notice that a block of 6 x 10 rectangles look fairly square. that leaves another rectangle, from which I can derive another block, say 6 x 4, or should that be 7 x 4? hard to tell. A "PQ" at this proportion, would be in the centre section of the ubiquitous 'Bell Curve'. As "P" & "Q" are selected by a random binary selection of 'fixed' length, the rim sector of the 'Bell Curve' would be either very square, or double-square, to begin with. 10 x 8, from this "P.Q" x 100 would make a double-square block.

My poor old vintage Cassio fx 201 P, (1978 model) can only cope with ten digits. My Fortlan....FortLan....as in Velly Pleased to meet you. Ploglam works fine so far. Now for the Big Numbers.....Boo Hoo, by tedious cut-n-paste. I need a Fortlan Ploglam to work with my Hewret Packard Notebook.

OK. hard to resist a few 'Fibbonacci Bunny Rabbits' I love the way those rectangles become square.

Those Bunnies are definitely Alice in Wonderland critters , changing size whichever way you look down the bunny- hole.

I shall have to spend a little time developing some drawing skills - I leave drawings and calculations all over the place , backs of envelopes,council tax bills . No wonder the Cat eyes me nervously !

Do you have to mate Casio and Hugh-splat Blagard ? It could produce a horrible child , especially with Fortlan as midwife. Give me a clue as to digit size limiting you Fortran progeny - is there some reason limiting it's capability . I get my baby basic to shuffle things about in arrays and ignore some of the preset commands which have limits.Set the individual digit free I say - Forget the Weatherfield one and release the swarm of Denary.Cry havoc ! and unleash the dogs of war (a bit literary for this time of day )

ps my desk buddy is a Casio fx-3900Pv (C1993 ) who's never had a new battery yet.

Last minute thought - If you're ever in sunny Leicester , the University have a little publicized garden (to keep out the hoi polloi ) which features a nice Fibonacci section (and no sun flowers in sight )

My mentor in Chaplaincy, The Rev. Roger Billings, now just retiring from St. John's in Burford (RAF Brize Norton http://en.wikipedia.org/wiki/RAF_Brize_Norton) comes from Leicester, I have many friends down that way. His Divine Grace Gaurangasundara Das for one. We jabber a way on the Dog & Bone in Latin. He keeps me up to scratch, as he was once a Roman Catholic Priest from Germany. If he offers you a Blessing, trust me, you are Blessed.

I last lived near Leicester around 1980 , but still have lots of family there . It's got a really vibrant mix of culture now . One of my brothers , much involved in 'happiness' (I add cryptically ) ,is based there - for further tease fun you may have read about him on a trans-Atlantic jaunt . I have 8 siblings in a multitude of places , which keeps life interesting. Just for the fun of it , another family figure hales from that place where 'look' sounds like 'Luke' - gave me hours of fun as a kid. I am something of a renegade , currently being in blue-bird territory ( funny , but I've never seen one ! Why on Earth did whoever dream up that one ? ).