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A102742
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Elite primes: a prime number p is called elite if only a finite number of Fermat numbers 2^(2^n)+1 are quadratic residues mod p.
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9
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3, 5, 7, 41, 15361, 23041, 26881, 61441, 87041, 163841, 544001, 604801, 6684673, 14172161, 159318017, 446960641, 1151139841, 3208642561, 38126223361, 108905103361, 171727482881, 318093312001, 443069456129, 912680550401, 1295536619521, 1825696645121, 2061584302081
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OFFSET
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1,1
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COMMENTS
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Let d = 2^r*d' be the multiplicative order of 2 modulo p. Note that 2^2^s == 2^d == 1 (mod p), so p divides none of.
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LINKS
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FORMULA
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PROG
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(PARI) list_upto(N)={forprime(p=3, N, r=2^valuation(p-1, 2); a=Mod(3, p); v=List(); k=0; while(1, listput(v, a); a=(a-1)^2+1; for(j=1, #v, if(v[j]==a, k=j; break(2)))); for(i=k, #v, znorder(v[i]) % r != 0 && next(2)); print1(p, ", "))} \\ Slow, only for illustration, Jeppe Stig Nielsen, Jan 28 2020
(PARI) isElite(n) = if(isprime(n) && n > 2, my(d = znorder(Mod(2, n)), StartPoint = valuation(d, 2), LengthTest = znorder(Mod(2, d >> StartPoint))); for(i = StartPoint, StartPoint + LengthTest - 1, if(issquare(Mod(2, n)^2^i + 1), return(0))); 1, 0) \\ Jianing Song, May 15 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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