A367014 Let q be the n-th prime power (A246655), then a(n) = q^3 + q^2 - q; number of solutions to x*y = z*w in the finite field F_q.

10, 33, 76, 145, 385, 568, 801, 1441, 2353, 4336, 5185, 7201, 12673, 16225, 20385, 25201, 30721, 33760, 51985, 70561, 81313, 105985, 120001, 151633, 208801, 230641, 266176, 305185, 362881, 394273, 499201, 537921, 578593, 712801, 921985, 1040401, 1103233, 1236385, 1306801

Offset

1,1

Comments

The number of solutions to x*y = z*w in F_q is Sum_{t in F_q} (number of solutions to x*y = t)^2. The number of solutions to x*y = 0 is 2*q-1, and the number of solutions to x*y = t for t != 0 is q-1, the number of units in F_q. So the total number is (2*q-1)^2 + (q-1)^2*(q-1) = q^3 + q^2 - q.

If q is odd, then a(n) is also the number of solutions to x^2 + y^2 = z^2 + w^2 in the finite field F_q.

Proof 1: the number is Sum_{t in F_q} (number of solutions to x^2 - z^2 = t)^2. For odd q, there is a one-to-one correspondence between the solutions to x*y = t and the solutions to x^2 - y^2 = t.

Proof 2: the number is Sum_{t in F_q} (number of solutions to x^2 + y^2 = t)^2. The number of solutions to x^2 + y^2 = 0 is 2*q-1 if q == 1 (mod 4) and 1 if q == 3 (mod 4), and the number of solutions to x^2 + y^2 = t for t != 0 is q-1 if q == 1 (mod 4) and q+1 if q == 3 (mod 4) (see A367013). So the total number is (2*q-1)^2 + (q-1)^2*(q-1) = q^3 + q^2 - q for q == 1 (mod 4) and 1^2 + (q+1)^2*(q-1) = q^3 + q^2 - q for q == 3 (mod 4).

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Example

For q = A246655(3) = 4, we see that in F_4 = F_2(t), where t^2 + t + 1 = 0:
 - x*y = z*w = 0 has 7 solutions for the pair (x,y) and 7 solutions for the pair (z,w);
 - x*y = z*w = 1 has 3 solutions for the pair (x,y) and 3 solutions for the pair (z,w);
 - x*y = z*w = t has 3 solutions for the pair (x,y) and 3 solutions for the pair (z,w);
 - x*y = z*w = 1+t has 3 solutions for the pair (x,y) and 3 solutions for the pair (z,w),
so a(4) = 7*7 + 3*3*3 = 76.

Mathematica

Map[#^3+#^2-#&, Select[Range[200], PrimePowerQ]] (* Paolo Xausa, Nov 26 2023 *)

Prog

(PARI) lim_A367014(N) = for(n=2, N, if(isprimepower(n), print1(n^3 + n^2 - n, ", ")))

Crossrefs

Cf. A246655, A101455 ({kronecker(-4,n)}).

Sequence in context: A299285 A081437 A085490 * A162433 A003012 A020478

Adjacent sequences: A367011 A367012 A367013 * A367015 A367016 A367017

Keyword

nonn

Author

Jianing Song, Nov 01 2023

Status

approved