Testwiki:Reference desk/Archives/Mathematics/2019 October 15

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October 15

Sum of four cubes of positive integers - conditions for divisibility by the sum of these numbers

By comparison to the previous section re the sum of three cubes of positive integers a, b, c, d, a similar question can be asked: What is the condition for the divisibility of this sum of four cubes by the sum of these numbers?--109.166.137.3 (talk) 10:30, 15 October 2019 (UTC)

Not sure if it's what you had in mind, but the identity analogous to the one above is:

$a^3+b^3+c^3+d^3=(a+b+c+d)(a^2+b^2+c^2+d^2-ab-ac-bc-ad-bd-cd)+3(abc+abd+acd+bcd),$

--RDBury (talk) 11:35, 15 October 2019 (UTC)

This is the searched identity. It appears that a sum of products appears, the number of products beeing determined by n choose k where k=3. How is this identity modified for the case of 5 cubes, 6 cubes.. and so on? (This determins the question below re generalization involving binomial coefficients).--109.166.137.3 (talk) 18:53, 16 October 2019 (UTC)

Largest prime with no Atkin–Goldwasser–Kilian–Morain certificate

What's the largest prime that doesn't have an Atkin–Goldwasser–Kilian–Morain certificate? 2 must not have one, since each AGKM certificate uses a smaller prime q; and for 3, 5 and 7, the lower bound on q works out larger than the previous prime. But does 11 have one? NeonMerlin 17:23, 15 October 2019 (UTC)