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May 15

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Can 2*12^n+1 be square for n>2?

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2*12^n+1 is square for n=1 and n=2, but can 2*12^n+1 be square again for n>2? 210.244.74.74 (talk) 21:04, 15 May 2023 (UTC)[reply]

It can't. Assume otherwise. Expand out the expression as , so that . Since , and , divides one term entirely.
If for , then . would imply that , which for can't be true by parity. So (and also by evenness, .) Divide both sides by to get . would once again lead to a problem, as would not make sense, so is even, implying that is odd, and thus . So . There is no integer for which this is the case, so we discard this case.
Now if for , then . would lead to , which would be impossible by parity, so . If , then , which cannot be for , so . Thus we can divide both sides by to get . once again would again lead to the impossible , so the right side is odd. Thus, and . This time, there are integer solutions to this, but they are . So we discard this case.
But since those are the only possible cases, there's a contradiction; so no has the property that is square. GalacticShoe (talk) 23:28, 15 May 2023 (UTC)[reply]

A problem of the Aliquot sum

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Let s(n) = sigma(n)-n = OEISA001065(n), s^k is the iterated function, we list the largest k such that a given natural number n is in the range of s^k.

k such natural numbers n OEIS sequence
0 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, … OEISA005114
1 208, 250, 362, 396, 412, 428, 438, 452, 478, 486, 494, 508, 672, 712, 716, 772, 844, 900, 906, 950, 1042, 1048, 1086, 1090, 1112, 1132, 1140, 1252, 1262, 1310, 1338, 1372, … OEISA283152
2 388, 606, 696, 790, 918, 1264, 1330, 1344, 1350, 1468, 1480, 1496, 1634, 1688, 1800, 1938, 1966, 2006, 2026, 2202, 2220, 2318, 2402, 2456, 2538, 2780, 2830, 2916, 2962, 2966, 2998, … OEISA284147

Can you find the sequence of k=infinity (i.e. n is in the range of s^k for all natural numbers k)? Assuming the strong version of Goldbach conjecture is true, i.e. all even number >6 are the sum of two distinct primes. 210.244.74.74 (talk) 21:15, 15 May 2023 (UTC)[reply]