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

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, 22, 18, 31, 6, 33, 26, 21, 21, 37, 38, 30, 9, 41, 42, 43, 33, 21, 46, 47, 9, 43, 22, 39, 30, 53, 38, 33, 21, 57, 44, 59, 27, 61, 62, 49, 11, 30, 66, 67

Offset

1,2

Comments

Conjecture: the only primes p for which a(p) < p are 5, 13, 17, 29. - Robert Israel, Jul 09 2017

Conjecture is true: see Math Overflow link. - Robert Israel, Apr 01 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Mathematics Overflow, Does the expression x^4+y^4 take on all values in Z/pZ (see answer by J. Silverman)

Example

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.:
   (0^4 + 0^4) mod 7 = ( 0 +  0) mod 7 =  0 mod 7 = 0;
   (0^4 + 1^4) mod 7 = ( 0 +  1) mod 7 =  1 mod 7 = 1;
   (1^4 + 1^4) mod 7 = ( 1 +  1) mod 7 =  2 mod 7 = 2;
   (1^4 + 2^4) mod 7 = ( 1 + 16) mod 7 = 17 mod 7 = 3;
   (2^4 + 2^4) mod 7 = (16 + 16) mod 7 = 32 mod 7 = 4;
   (1^4 + 3^4) mod 7 = ( 1 + 81) mod 7 = 82 mod 7 = 5;
   (2^4 + 3^4) mod 7 = (16 + 81) mod 7 = 97 mod 7 = 6.
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.)

Maple

f1:= proc(n) option remember; local S;
    S:= {seq(x^4 mod n, x=0..n-1)};
  nops({seq(seq(S[i]+S[j] mod n, i=1..j), j=1..nops(S))});
end proc:
f:= proc(n) local t;
mul(f1(t[1]^t[2]), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jul 09 2017

Mathematica

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];
f[n_] := Module[{p, e}, Product[{p, e} = pe; f1[p^e], {pe, FactorInteger[n]}]];
Array[f, 100] (* Jean-François Alcover, Jul 30 2020, after Maple *)

Prog

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

Crossrefs

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

Sequence in context: A205442 A049982 A245642 * A070167 A168113 A170895

Adjacent sequences: A289556 A289557 A289558 * A289560 A289561 A289562

Keyword

nonn,mult

Author

Jon E. Schoenfield, Jul 08 2017

Status

approved