

A367014


Let q be the nth 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.


1



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
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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*q1, and the number of solutions to x*y = t for t != 0 is q1, the number of units in F_q. So the total number is (2*q1)^2 + (q1)^2*(q1) = 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 onetoone 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*q1 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 q1 if q == 1 (mod 4) and q+1 if q == 3 (mod 4) (see A367013). So the total number is (2*q1)^2 + (q1)^2*(q1) = q^3 + q^2  q for q == 1 (mod 4) and 1^2 + (q+1)^2*(q1) = q^3 + q^2  q for q == 3 (mod 4).


LINKS



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



KEYWORD

nonn


AUTHOR



STATUS

approved



