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A158470 Numbers k such that prime(k-1) + 7 is square and equal to prime(k+1) - 1. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=2..28.

Cino Hilliard, C program (broken link) [From Cino Hilliard, Mar 21 2009]

S. M. Ruiz, Integer equal

Sebastian Martin Ruiz and others, Integers then Equals, digest of 7 messages in primenumbers Yahoo group, Mar 14 - Mar 20, 2009.

Zak Seidov, A158470 First 100 terms

EXAMPLE

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.

MATHEMATICA

pnsQ[n_]:=Module[{p1=Prime[n-1], p2=Prime[n+1]}, p1+7==p2-1&&IntegerQ[ Sqrt[ p1+7]]]; Select[Range[5, 25660000], pnsQ]

PROG

(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))

)

)

)

}

CROSSREFS

Sequence in context: A173851 A358340 A295840 * A163933 A359987 A099839

Adjacent sequences: A158467 A158468 A158469 * A158471 A158472 A158473

KEYWORD

nonn

AUTHOR

Cino Hilliard, Mar 19 2009

EXTENSIONS

More terms from Zak Seidov, Mar 20 2009

STATUS

approved

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Last modified January 29 01:41 EST 2023. Contains 359905 sequences. (Running on oeis4.)