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A272069
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Odd numbers n such that 3^n+1 is a sum of two squares.
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0
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1, 5, 13, 65, 149, 281, 409, 421, 449, 461, 577, 761
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Keenan Curtis, "Sums of Two Squares: An Analysis of Numbers of the form 2^n+1 and 3^n+1", submitted to INVOLVE.
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LINKS
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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.
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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a(12) = 761 added from the Cunningham Project via Greg Dresden, Jul 23 2016
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STATUS
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approved
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