

A272069


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


0



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



