A272069 Odd numbers n such that 3^n+1 is a sum of two squares.

1, 5, 13, 65, 149, 281, 409, 421, 449, 461, 577, 761

Offset

1,2

Comments

Keenan Curtis found the values up though 577 in his undergraduate thesis (working with Jeremy Rouse at Wake Forest University). Keenan proved that if 3^n+1 is a sum of two squares for n odd, then n must be equivalent to 1 mod 4, that n itself is a sum of two squares, and that 3^p+1 is a sum of two squares for all primes p dividing n.

1289, 1741, 1913, 1993, 5081, 9209, 11257, 13093, 14957, 26633, 45553, 60917, 81761 are terms. Except for 1 and 65, all terms so far are prime. - Chai Wah Wu, Jul 23 2020

References

Keenan Curtis, "Sums of Two Squares: An Analysis of Numbers of the form 2^n+1 and 3^n+1", submitted to INVOLVE.

Links

Table of n, a(n) for n=1..12.

Greg Dresden, Kylie Hess, Saimon Islam, Jeremy Rouse, Aaron Schmitt, Emily Stamm, Terrin Warren, Pan Yue, When is a^n+1 the sum of two squares?, arXiv:1609.04391 [math.NT], 2016. See p. 20.

S. S. Wagstaff, Jr., The Cunningham Project

Example

3^1+1 = 4 = 0^2 + 2^2, so 1 is a term;
3^5+1 = 244 = 10^2 + 12^2, so 5 is a term;
3^13+1 = 1594324 = 82^2 + 1260^2, so 13 is a term.

Crossrefs

Intersection of A000404 and A034472.

Sequence in context: A149574 A301634 A309167 * A018678 A149575 A156101

Adjacent sequences: A272066 A272067 A272068 * A272070 A272071 A272072

Keyword

nonn,hard,more

Author

Greg Dresden, Apr 19 2016

Extensions

a(12) = 761 added from the Cunningham Project via Greg Dresden, Jul 23 2016

Status

approved