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A038703
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Primes p such that p^2 mod q is odd, where q is the previous prime.
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1
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OFFSET
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1,1
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COMMENTS
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The next term if it exists is > 32452843 = 2000000th prime. Can someone prove this sequence is complete? - Olivier Gérard, Jun 26 2001
To prove that 127 is the last prime, we need to show that prime gaps satisfy prime(k)-prime(k-1)<sqrt(prime(k-1)) for k>31. Although it is easy to verify this inequality for all known prime gaps, there is no proof for all gaps. - T. D. Noe, Jul 25 2006
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LINKS
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FORMULA
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Prime(k) is in the sequence if prime(k)^2 (mod prime(k-1)) is odd.
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EXAMPLE
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The first prime with a prime lower than itself is 3. This squared is 9, which when divided by the previous prime 2 leaves remainder 1, which is odd. So 3 is in the sequence. 11 is not in the sequence because 11^2, when divided by the previous prime 7, leaves a remainder of 121 (mod 7) = 2, which is even.
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MATHEMATICA
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Prime /@ Select[ Range[ 2, 100 ], OddQ[ Mod[ Prime[ # ]^2, Prime[ # - 1 ] ] ] & ]
Transpose[Select[Partition[Prime[Range[50]], 2, 1], OddQ[PowerMod[Last[#], 2, First[#]]]&]] [[2]] (* Harvey P. Dale, May 31 2012 *)
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PROG
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(PARI) isok(p) = isprime(p) && (p>2) && (lift(Mod(p, precprime(p-1))^2) % 2); \\ Michel Marcus, Mar 05 2023
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CROSSREFS
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Cf. A058188 (number of primes between prime(n) and prime(n)+sqrt(prime(n))).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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