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A145824
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Lower twin primes p1 such that p1-1 is a square.
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4
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5, 17, 101, 197, 5477, 8837, 16901, 17957, 21317, 25601, 52901, 65537, 106277, 115601, 122501, 164837, 184901, 193601, 220901, 341057, 401957, 470597, 490001, 495617, 614657, 739601, 846401, 972197, 1110917, 1144901, 1336337, 1464101
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OFFSET
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1,1
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COMMENTS
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3 is the only lower twin prime for which p1+1 is a square. This follows from the fact that lower twin primes > 3 are of the form 3n+2 and adding 1 we get a number of the form 3m. Then 3m = k^2 implies k = 3r and 3m = 9r^2. Subtracting 1 we have 3m = (3r-1)(3r+1) not prime contradicting 3m-1 is prime.
Conjecture: There exist an infinite number of primes of this form.
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LINKS
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EXAMPLE
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p1 = 5 is a lower twin prime. 5-1 = 4 is a square.
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MATHEMATICA
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PROG
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(PARI) g(n) = for(x=1, n, y=twinl(x)-1; if(issquare(y), print1(y+1", ")))
twinl(n) = local(c, x); c=0; x=1; while(c<n, if(ispseudoprime(prime(x)+2), c++);
x++; ); return(prime(x-1))
(Magma) [p: p in PrimesUpTo(2000000) | IsSquare(p-1) and IsPrime(p+2)]; // Vincenzo Librandi, Nov 08 2014
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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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