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A285282
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Numbers n such that n^2 + 1 is 13-smooth.
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3
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OFFSET
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1,2
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COMMENTS
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Equivalently: Numbers n such that all prime factors of n^2 + 1 are <= 13.
Since an odd prime factor of n^2 + 1 must be of the form 4m + 1, n^2 + 1 must be of the form 2*5^a*13^b.
This sequence is complete by a theorem of Størmer.
The largest instance 239^2 + 1 = 2*13^4 also gives the only nontrivial solution for x^2 + 1 = 2y^4 (Ljunggren).
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REFERENCES
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W. Ljunggren, Zur Theorie der Gleichung x^2 + 1 = 2y^4, Avh. Norsk Vid. Akad. Oslo. 1(5) (1942), 1--27.
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LINKS
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EXAMPLE
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For n = 8, a(8)^2 + 1 = 57^2 + 1 = 3250 = 2*5^3*13.
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MATHEMATICA
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PROG
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(PARI) for(n=1, 9e6, if(vecmax(factor(n^2+1)[, 1])<=13, print1(n", ")))
(Python)
from sympy import primefactors
def ok(n): return max(primefactors(n**2 + 1))<=13 # Indranil Ghosh, Apr 16 2017
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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