%I #20 Jul 30 2020 07:17:39

%S 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 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 53,38,33,21,57,44,59,27,61,62,49,11,30,66,67

%N Number of modulo n residues among sums of two fourth powers.

%C Conjecture: the only primes p for which a(p) < p are 5, 13, 17, 29. - _Robert Israel_, Jul 09 2017

%C Conjecture is true: see Math Overflow link. - _Robert Israel_, Apr 01 2020

%H Robert Israel, <a href="/A289559/b289559.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 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 (0^4 + 0^4) mod 7 = ( 0 + 0) mod 7 = 0 mod 7 = 0;

%e (0^4 + 1^4) mod 7 = ( 0 + 1) mod 7 = 1 mod 7 = 1;

%e (1^4 + 1^4) mod 7 = ( 1 + 1) mod 7 = 2 mod 7 = 2;

%e (1^4 + 2^4) mod 7 = ( 1 + 16) mod 7 = 17 mod 7 = 3;

%e (2^4 + 2^4) mod 7 = (16 + 16) mod 7 = 32 mod 7 = 4;

%e (1^4 + 3^4) mod 7 = ( 1 + 81) mod 7 = 82 mod 7 = 5;

%e (2^4 + 3^4) mod 7 = (16 + 81) mod 7 = 97 mod 7 = 6.

%e 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 f1:= proc(n) option remember; local S;

%p S:= {seq(x^4 mod n, x=0..n-1)};

%p nops({seq(seq(S[i]+S[j] mod n,i=1..j),j=1..nops(S))});

%p end proc:

%p f:= proc(n) local t;

%p mul(f1(t[1]^t[2]), t = ifactors(n)[2])

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jul 09 2017

%t 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 f[n_] := Module[{p, e}, Product[{p, e} = pe; f1[p^e], {pe, FactorInteger[n]}]];

%t Array[f, 100] (* _Jean-François Alcover_, Jul 30 2020, after Maple *)

%o (PARI) a(n) = #Set(vector(n^2, i, ((i%n)^4 + (i\n)^4) % n)); \\ _Michel Marcus_, Jul 08 2017

%Y Cf. A155918 (gives number of modulo n residues among sums of two squares).

%K nonn,mult

%O 1,2

%A _Jon E. Schoenfield_, Jul 08 2017