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A158470
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Numbers k such that prime(k-1) + 7 is square and equal to prime(k+1) - 1.
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3
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11, 105, 210, 4352, 13631, 171030, 206287, 482817, 507376, 669211, 1043655, 1347091, 2078002, 3272095, 3372558, 3433588, 3551781, 6584471, 6738010, 7186808, 7604542, 8426927, 10893207, 13200411, 15175773, 23350193, 25653343
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OFFSET
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1,1
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COMMENTS
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If the condition holds, prime(k-1) and prime(k) are twin primes. These are of the form 10m+9 and 10m+1. I.e., the last digits are 9 and 1. This is true because a square number must end in 0,1,4,5,6,9. So prime(k-1)+7 is square => it must end in one of these numbers. So to find the ending of prime(k-1), we subtract 7 from 0,1,4,5,6,9 to get the last digit, i.e., 3,4,7,8,9,2.
Since prime(k-1) is prime, endings 2,4,5,8 are not allowed. This leaves us with 3,7,9 as possible endings of prime(k-1). Now to get prime(k), which the condition states is 2 greater than prime(k-1), we add 2 and 3+2=5 => prime(k) not prime, impossible. So the possible endings of prime(k-1) are reduced to 7 or 9. Now the condition prime(k-1)+7 = prime(k+1)-1 => prime(k-1)+8 = prime(k+1). Then adding 7 => prime(k+1) ends in 5, impossible. So prime(k-1) must end in 9, and adding 2 makes prime(k) end in 1. This sequence is a calculation of the conjecture provided in the link. The PARI script provides for the general investigation of numbers of the form prime(k-1)+a and prime(k+1)-b. The values a=5,7; b=1 consistently yield twin primes when the condition holds.
Notice we test for square of the first prime(k-1) retrieval before calling the second prime(k+1). This cuts the search time in half. A far superior search routine is the C program found in the link, which reads a huge 300 GB file of the primes < 1 trillion + 1 billion.
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LINKS
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Sebastian Martin Ruiz and others, Integers then Equals, digest of 7 messages in primenumbers Yahoo group, Mar 14 - Mar 20, 2009.
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EXAMPLE
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For k=11, prime(11-1) = 29, 29+7 = 36 = prime(11+1)-1 = 37-1 so 11 is the first entry in the sequence.
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MATHEMATICA
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pnsQ[n_]:=Module[{p1=Prime[n-1], p2=Prime[n+1]}, p1+7==p2-1&&IntegerQ[ Sqrt[ p1+7]]]; Select[Range[5, 25660000], pnsQ]
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PROG
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(PARI) integerequal(m, n, a, b) =
{
local(x, p1, p2);
for(x=m, n,
p1=prime(x-1);
if(issquare(p1+a),
p2=prime(x+1); if((p1+a)==(p2-b),
print(x", "p1", "prime(x))
)
)
)
}
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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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