A few years ago I 'lost' a week - solving this. I'll post the answers, if needed, after 1 week.

Twin primes : Prime numbers differing by 2.
Mersenne Prime : 2n – 1 where n is prime.
Triangular numbers : n(n + 1)/2 where n is ordinal.
Pythagorean triples : set of 3 whole numbers, with no common factor, forming the sides of a right-angled triangle.
Sequence : each number after the second is the sum of the previous two, starting with 2,7,9,16.......
No entry begins with '0'. Squares, cubes and higher powers are always
Editorial Note: Questions Revised 04/25/08 at 2:20 PM
Across
1 Number from the sequence.
3 Pythagorean triple with 3d and 30d.
6 Twin prime with 30d.
7 Triangular number, see 13d.
8 Number from the sequence.
12 2d squared.
13 Cube of a square.
15 11d squared.
17 Ordinal for 14d.
20 Ordinal for 33a.
22 Cube of a prime.
25 6a cubed.
26 Pythagorean triple with 29a and 22d.
28 Triangular number, see 31a.
29 Pythagorean triple with 26a and 22d.
31 Mersenne prime, also ordinal for 28a.
32 Reverse of 29d.
33 Triangular number, see 20a.
DOWN
1 2d cubed.
2 Twin prime with 4d.
3 Pythagorean triple with 3a and 30d.
4 See 2d.
5 The ordinal for 27d.
7 Product of a 4th power and a prime.
9 Product of a cube and a prime.
10 Mersenne prime.
11 6a squared.
13 Ordinal for 7a.
14 Triangular number, see 17a.
16 A cube.
18 Product of a cube and a prime.
19 Product of a 5th power and a prime.
21 Number from the sequence.
22 Pythagorean triple with 26a and 29a
23 Prime.
24 Product of a prime cubed and a prime squared.
27 Triangular number, see 5d.
28 Square.
29 Prime.
30 See 6a, also Pythagorean triple with 3d and 3a.
31 A 5th power.