OFFSET
1,1
COMMENTS
By Hasse's theorem, every elliptic curve E over GF(q) has cardinality at most q + 1 + floor(2*sqrt(q)). Moreover, for every prime power q, there exists an elliptic curve E over GF(q) with cardinality at least q + floor(2*sqrt(q)). Thus these are the prime powers q for which A005523(n) = q + floor(2*sqrt(q)), where q = A246655(n).
By a theorem of Deuring and Waterhouse, these are exactly the prime powers q = p^k such that q is not prime, q is not a square, and p divides floor(2*sqrt(q)).
LINKS
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
EXAMPLE
The first few values of the sequence (factorized) are 2^7, 2^11, 3^7, 7^5, 2^15, 2^17, 2^19, 5^9, 2^21, 2^23, 3^15, 5^11, 2^27, 2^29, ...
PROG
(Sage)
for q in range(1, 100000):
if Integer(q).is_prime_power():
p = Integer(q).prime_factors()[0]
if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):
print(q)
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Aug 02 2023
STATUS
approved