Date: Apr 22, 2013 2:31 AM
Author: David Bernier
Subject: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
I was looking for a simple arithmetic sequence with

many primes "crowded together", i.e. quasi-consecutive ...

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.