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 A321616 Primes p = k^2 + (k-1)^2 such that k^p - (k-1)^p is prime. 0
 5, 61, 113, 1741 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Conjecture: generally, these are primes p = a^2 + b^2 with a > b > 0 such that (a^p - b^p)/(a-b) is prime, so must be a-b = 1. It seems that there are no primes (a^q + b^q)/(a+b) for primes q = a^2 + b^2 > 5. Especially, there are probably no primes q = m^2 + 1 > 5 such that (m^q - 1)/(m-1) is prime or (m^q + 1)/(m+1) is prime. How to prove it? No more terms up to the prime 19801 = 100^2 + 99^2. - Amiram Eldar, Nov 15 2018 LINKS EXAMPLE The prime 5 = 2^2 + 1^2 and 2^5 - 1^5 = 31 is prime. We have 61 = 6^2 + 5^2, 113 = 8^2 + 7^2, 1741 = 30^2 + 29^2. MATHEMATICA f[k_]:=k^2 + (k-1)^2 ; seqQ[k_]:=Module[{p=f[k]}, PrimeQ[p] && PrimeQ[k^p - (k-1)^p ]]; f[Select[Range, seqQ]] (* Amiram Eldar, Nov 15 2018 *) PROG (PARI) lista(nn) = {for (k=1, nn, if (isprime(p=k^2 + (k-1)^2) && isprime(k^p - (k-1)^p), print1(p, ", ")); ); } \\ Michel Marcus, Nov 18 2018 CROSSREFS Cf. A002144, A121091. Subsequence of A027862. Sequence in context: A139915 A174053 A107191 * A182352 A142643 A201848 Adjacent sequences:  A321613 A321614 A321615 * A321617 A321618 A321619 KEYWORD nonn,more AUTHOR Thomas Ordowski, Nov 15 2018 STATUS approved

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Last modified February 25 21:55 EST 2020. Contains 332264 sequences. (Running on oeis4.)