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A289559 Number of modulo n residues among sums of two fourth powers.

%I A289559 #20 Jul 30 2020 07:17:39
%S A289559 1,2,3,3,3,6,7,3,7,6,11,9,10,14,9,3,13,14,19,9,21,22,23,9,11,20,19,21,
%T A289559 22,18,31,6,33,26,21,21,37,38,30,9,41,42,43,33,21,46,47,9,43,22,39,30,
%U A289559 53,38,33,21,57,44,59,27,61,62,49,11,30,66,67
%N A289559 Number of modulo n residues among sums of two fourth powers.
%C A289559 Conjecture: the only primes p for which a(p) < p are 5, 13, 17, 29. - _Robert Israel_, Jul 09 2017
%C A289559 Conjecture is true: see Math Overflow link. - _Robert Israel_, Apr 01 2020
%H A289559 Robert Israel, <a href="/A289559/b289559.txt">Table of n, a(n) for n = 1..10000</a>
%H A289559 Mathematics Overflow, <a href="https://mathoverflow.net/questions/356270/does-the-expression-x4-y4-take-on-all-values-in-mathbbz-p-mathbbz">Does the expression x^4+y^4 take on all values in Z/pZ</a> (see answer by J. Silverman)
%e A289559 a(7) = 7 because (j^4 + k^4) mod 7, where j and k are integers, can take on all 7 values 0..6; e.g.:
%e A289559    (0^4 + 0^4) mod 7 = ( 0 +  0) mod 7 =  0 mod 7 = 0;
%e A289559    (0^4 + 1^4) mod 7 = ( 0 +  1) mod 7 =  1 mod 7 = 1;
%e A289559    (1^4 + 1^4) mod 7 = ( 1 +  1) mod 7 =  2 mod 7 = 2;
%e A289559    (1^4 + 2^4) mod 7 = ( 1 + 16) mod 7 = 17 mod 7 = 3;
%e A289559    (2^4 + 2^4) mod 7 = (16 + 16) mod 7 = 32 mod 7 = 4;
%e A289559    (1^4 + 3^4) mod 7 = ( 1 + 81) mod 7 = 82 mod 7 = 5;
%e A289559    (2^4 + 3^4) mod 7 = (16 + 81) mod 7 = 97 mod 7 = 6.
%e A289559 a(16) = 3 because (j^4 + k^4) mod 16 can take on only the three values 0, 1, and 2. (This is because j^4 mod 16 = 0 for all even j and 1 for all odd j.)
%p A289559 f1:= proc(n) option remember; local S;
%p A289559     S:= {seq(x^4 mod n, x=0..n-1)};
%p A289559   nops({seq(seq(S[i]+S[j] mod n,i=1..j),j=1..nops(S))});
%p A289559 end proc:
%p A289559 f:= proc(n) local t;
%p A289559 mul(f1(t[1]^t[2]), t = ifactors(n)[2])
%p A289559 end proc:
%p A289559 map(f, [$1..100]); # _Robert Israel_, Jul 09 2017
%t A289559 f1[n_] := f1[n] = Module[{S = Table[Mod[x^4, n], {x, 0, n-1}] // Union}, Table[Mod[S[[i]] + S[[j]], n], {j, 1, Length[S]}, {i, 1, j}] // Flatten // Union // Length];
%t A289559 f[n_] := Module[{p, e}, Product[{p, e} = pe; f1[p^e], {pe, FactorInteger[n]}]];
%t A289559 Array[f, 100] (* _Jean-François Alcover_, Jul 30 2020, after Maple *)
%o A289559 (PARI) a(n) = #Set(vector(n^2, i, ((i%n)^4 + (i\n)^4) % n)); \\ _Michel Marcus_, Jul 08 2017
%Y A289559 Cf. A155918 (gives number of modulo n residues among sums of two squares).
%K A289559 nonn,mult
%O A289559 1,2
%A A289559 _Jon E. Schoenfield_, Jul 08 2017