Squarejacks

Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit,
together with the prime limit.

2 2
3 3
4 2
5 5
6 3
7 7
8 2
9 3
10 5
11 11
12 3
13 13
14 7
15 5
16 2
17 17
18 3
19 19
20 5
21 7
22 11
23 23
24 3
25 5
26 13
27 3
33 11
34 17
35 7
39 13
45 5
49 7
50 5
51 17
55 11
56 7
64 2
65 13
69 23
76 19
77 11
91 13
99 11
120 5
153 17
161 23
169 13
170 17
208 13
209 19
323 19
324 3
351 13
391 23
441 7
2024 23
2431 17

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit,
> together with the prime limit.

Sorry, my "prime limit" rountine is actually calculating the number of
primes which appear in the factorization.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit,
> together with the prime limit.

2 3
3 3
4 5
5 5
6 7
7 7
8 7
9 5
10 11
11 11
12 13
13 13
14 13
15 7
16 17
17 17
18 19
19 19
20 19
21 11
22 23
23 23
24 23
25 13
26 13
27 13
33 17
34 17
35 17
39 19
45 23
49 7
50 17
51 17
55 11
56 19
64 13
65 13
69 23
76 19
77 19
91 23
99 11
120 17
153 19
161 23
169 17
170 19
208 23
209 19
323 23
324 19
351 13
391 23
441 17
2024 23
2431 19

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >>Here is a list n < 5000 such that n^2/(n^2-1) is within the 23 limit,
>>together with the prime limit.

That's more like it! I can't find any more with n<100000, so it might be complete.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:

> That's more like it! I can't find any more with n<100000, so it
might
> be complete.

Speaking of complete, now that Catalan's conjecture (now Mihailescu's
theorem) has finally been proven we know that 9/8 is the only
superparticular ratio where the numerator and denominator are both
powers. The proof uses a lot of deep algebraic number theory. These
sorts of things are tough to prove!