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 A260911 Least positive integer k < prime(n) such that there are 0 < i < j < k for which i^2 + j^2 = k^2 and i,j,k are all quadratic residues modulo prime(n), or 0 if no such k exists. 5
 0, 0, 0, 0, 5, 0, 0, 0, 0, 25, 0, 34, 0, 41, 25, 25, 5, 5, 26, 5, 37, 0, 41, 0, 0, 65, 17, 34, 5, 61, 17, 5, 17, 25, 25, 29, 37, 26, 25, 41, 5, 5, 5, 25, 25, 53, 34, 17, 34, 5, 109, 5, 5, 5, 17, 37, 34, 41, 34, 53 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: (i) a(n) > 0 for all n > 25. In other words, for any prime p > 100, we have a^2 + b^2 = c^2 for some a,b,c in the set R(p) = {0 50, we have a^2 + b^2 = c^2 for some a,b,c in the set N(p) = {0 32, we have a^2 + b^2 = c^2 for some a,b in the set R(p) and c in the set N(p). (iv) For any prime p > 72, we have a^2 + b^2 = c^2 for some a,b in the set N(p) and c in the set R(p). I have verified the conjecture for primes p < 1.5*10^7. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..10000 EXAMPLE a(10) = 25 since 7^2 + 24^2 = 25^2, and 7, 24, 25 are all quadratic residues modulo prime(10) = 29. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] Do[Do[If[JacobiSymbol[k, Prime[n]]<1, Goto[bb]]; Do[If[JacobiSymbol[j, Prime[n]]<1, Goto[cc]]; If[SQ[k^2-j^2]&&JacobiSymbol[Sqrt[k^2-j^2], Prime[n]]==1, Print[n, " ", k]; Goto[aa]]; Label[cc]; Continue, {j, 1, k-1}]; Label[bb]; Continue, {k, 1, Prime[n]-1}]; Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 50}] CROSSREFS Cf. A000040, A000290, A257364. Sequence in context: A270030 A284104 A369732 * A228631 A101194 A196344 Adjacent sequences: A260908 A260909 A260910 * A260912 A260913 A260914 KEYWORD nonn AUTHOR Zhi-Wei Sun, Aug 03 2015 STATUS approved

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Last modified September 18 20:35 EDT 2024. Contains 376002 sequences. (Running on oeis4.)