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primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
Posted:
Apr 22, 2013 2:31 AM


I was looking for a simple arithmetic sequence with many primes "crowded together", i.e. quasiconsecutive ...
Suppose we let n = 1,097,495,500,000 ; then I get this:
n+19941 is prime, n+19971 is prime, n+20001 is prime, n+20031 is prime, n+20061 is prime, n+20091 is prime,
n+20121 is composite,
n+20151 is prime, n+20181 is prime, n+20211 is prime, n+20241 is prime, n+20271 is prime, n+20301 is composite.
1,097,495,520,121 = 7*11*13*23*47669527 // n+20121
1,097,495,520,301 = 61*27617*651473. // n+20301
So, it should be possible to have a block of six consecutive numbers from the arithmectic sequence: 1, 31, 61, 91, 121, ... 30k+1 ...
that are all prime, then a composite number, followed by a second block of six consecutive numbers from that arithmetic sequence that are all prime ... (probably?)
I.e. Out of thirteen consecutive integers from the sequence of the 30k+1, can we get at least 12 primes out of the thirteen numbers, for the right choice of the 13 consecutive numbers ?
David Bernier
? isprime(n+19941) %37 = 1 ? isprime(n+19971) %38 = 1 ? isprime(n+20001) %39 = 1 ? isprime(n+20031) %47 = 1 ? isprime(n+20061) %48 = 1 ? isprime(n+20091) %49 = 1 ? isprime(n+20121) %40 = 0 ? isprime(n+20151) %41 = 1 ? isprime(n+20181) %42 = 1 ? isprime(n+20211) %43 = 1 ? isprime(n+20241) %44 = 1 ? isprime(n+20271) %45 = 1 ? isprime(n+20301) %46 = 0
 Jesus is an Anarchist.  J.R.



