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Topic: primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
Replies: 4   Last Post: Apr 24, 2013 11:24 AM

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David Bernier

Posts: 3,369
Registered: 12/13/04
primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...
Posted: Apr 22, 2013 2:31 AM
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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.



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