

A062775


Number of Pythagorean triples mod n: total number of solutions to x^2 + y^2 = z^2 mod n.


7



1, 4, 9, 24, 25, 36, 49, 96, 99, 100, 121, 216, 169, 196, 225, 448, 289, 396, 361, 600, 441, 484, 529, 864, 725, 676, 891, 1176, 841, 900, 961, 1792, 1089, 1156, 1225, 2376, 1369, 1444, 1521, 2400, 1681, 1764, 1849, 2904, 2475, 2116, 2209, 4032, 2695, 2900
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OFFSET

1,2


COMMENTS

a(n) is multiplicative and, for a prime p, a(p) = p^2. Hence a(n) = n^2 if n is squarefree.


REFERENCES

See newsgroup sci.math.research; subject: Re: Pythagorean triples mod n / Solution enhanced; author: Gottfried Helms; MessageID: brv5i2$28v$1(AT)news.ks.uiuc.edu; Date: Fri, Dec 19 2003 15: 30: 10 +0000 (UTC)


LINKS

T. D. Noe, Table of n, a(n) for n=1..1000
L. Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214, 2014


FORMULA

a(n) is multiplicative. For the powers of primes p, there are four cases. For p=2, there are cases for even and odd powers: a(2^(2k1)) = 2^(3k1) (2^k1) and a(2^(2k)) = 2^(3k) (2^(k+1)1). Similarly, for odd primes p, a(p^(2k1)) = p^(3k2) (p^k+p^(k1)1) and a(p^(2k)) = p^(3k1) (p^(k+1)+p^k1).  T. D. Noe, Dec 22 2003
Thanks to T. D. Noe and Gottfried Helms for additional comments, Dec 23, 2003.
If the canonical form of n is n = 2^i*3^j*5^k*...p^q then it appears that a(n) = n*f(2, i)*f(3, j)*f(5, k)*...*f(p, q) where f(p, 1) = p for any prime p; f(2, i) = 2^i + 2^i  2^ceil(i/2); f(p, i) = p^i + p^(i1)  p^floor((i1)/2) for any odd prime p. For example a(7) = 49 because a(7) = 7*f(7, 1) = 7*7; a(16) = 448 because a(16) = a(2^4)= 16 * f(2, 4) = 16 * (16+164) = 16*28 = 448; a(12) = 216 because a(12) = a(3*2^2)= 12*f(2, 2)*f(3, 1) = 12*(4+42)*3 = 216.  Gottfried Helms, May 13 2004


MATHEMATICA

Table[cnt=0; Do[If[Mod[x^2+y^2z^2, n]==0, cnt++ ], {x, 0, n1}, {y, 0, n1}, {z, 0, n1}]; cnt, {n, 50}]


CROSSREFS

Cf. A091143 (number of solutions to x^2 + y^2 = z^2 mod 2^n).
Cf. A060968, A063454.
Sequence in context: A270685 A272252 A067801 * A270450 A270461 A046422
Adjacent sequences: A062772 A062773 A062774 * A062776 A062777 A062778


KEYWORD

nonn,nice,mult


AUTHOR

Ahmed Fares (ahmedfares(AT)mydeja.com), Jul 18 2001


EXTENSIONS

More terms from Sascha Kurz, Mar 25 2002


STATUS

approved



