primes in the arithmetic sequence 1, 31, 61, 91, 121, ... 30k+1 ...

[David Bernier]

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.

[Don Reble]

> Out of thirteen consecutive [30k+1] integers, can we get at least 12 primes...?

389232355162471 prime
389232355162501 prime
389232355162531 prime
389232355162561 prime
389232355162591 prime
389232355162621 prime
389232355162651 = 7 7 11 13 23 23609 102299
389232355162681 prime
389232355162711 prime
389232355162741 prime
389232355162771 prime
389232355162801 prime
389232355162831 prime

The middle number must be a multiple of 7 (otherwise there would be
two multiples of 7), and there must be multiples of 11 and 13, all
the same number. So the middle number is 11011 modulo 30030. That
hastens the search.

--
Don Reble d...@nk.ca

[christian.bau]

> 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 ?

Of those 13 consecutive numbers, one or two are divisible by 7; one or
two are divisible by 11, one is divisible by 13, at most one divisible
by 17 etc. To have only one divisible by 7, it must be the middle one.
To have only one number composite, that number must also be divisible
by 11 and 13. 1001 = 7x11x13. So you need to check

(1001 * (30k + 11)) - 180, -150, -120, -90, -60, -30, +30, +60,
+90, +120, +150, +180.

389,232,355,162,471 + 0, 30, 60, 90, 120, 150, 210, 240, 270, 300,
330, 360 are all primes.

[David Bernier]

I was enthused that two people (Don Reble and yourself)
found examples of what I was looking for and posted.


If S = {0,30,60,90,120,150,210,240,270,300,330,360}
then if the prime p is set to p=2,
none of the numbers in S+1 is congruent to 0 (mod 2).

If p=3,
none of x in S+1 is congruent to 0 (mod 3).
If p=5
(same with S+1)

If p=7, none of the x in S+2 is congruent to 0 (mod 7).

If p=11, none of the x in S+5 is congruent to 0 (mod 11)

if p>11 is a prime, there exits n_p such that
if x is in S, then x + n_p == 0 (mod p)
(Note: if p> 361, this is not hard to see).

So, there's no modular arithmetic "obstruction"
to the existence of C>0 such that for all
x in S, x+C is a prime.

With C = 389,232,355,162,471
from your computations, all the numbers x+C for
x in S = {0,30,60,90,120,150,210,240,270,300,330,360}
are prime.

S could be called a "translated set of primes" candidate,
for instance.

In the same way, if S_2 = {0, 2}, S_2 is a
"translated set of primes" candidate: connected to
twin primes.

The set S has a maximum difference between elements of
360, and has a cardinality of 12.

I'm wondering how large in cardinality
a "translated set of primes" candidate set T can be
if, say, the maximum difference between elements of T is
at most 360.

It's a way of looking at potential primes clumpiness over "small"
distances.

David Bernier

[Don Reble]

> I'm wondering how large in cardinality
> a "translated set of primes" candidate set T can be
> if, say, the maximum difference between elements of T is
> at most 360.
>
> It's a way of looking at potential primes clumpiness over "small"
> distances.

Search for "prime constellation" and "prime k-tuple".

--
Don Reble d...@nk.ca

[Gerard Schildberger]

... also, metric primes. ______________________ Gerard Schildberger