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## Patterns in prime numbers (themselves, not their frequency)

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• I m interested in the idea that a number can be ruled out as a prime number due to patterns occuring within the digits of that number. For example, you might
Message 1 of 4 , Aug 4, 2004
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I'm interested in the idea that a number can be ruled out as a prime
number due to patterns occuring within the digits of that number.
For example, you might say, "a number containing five nines in a row
can't (or likely won't) be a prime number, for this reason: ...",
or "any number containing the sequence '1-2-3-4' followed by that
same sequence again won't be prime, because: ..."

Are there any rules which describe prime numbers in such a way?
• ... No, let N be any number then there is a prime containing a consecutive subsequence of digits equal to N. One way to show this is as follows: Let N =
Message 2 of 4 , Aug 4, 2004
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On Wed, 4 Aug 2004, isbat1 wrote:

> I'm interested in the idea that a number can be ruled out as a prime
> number due to patterns occuring within the digits of that number.
> For example, you might say, "a number containing five nines in a row
> can't (or likely won't) be a prime number, for this reason: ...",
> or "any number containing the sequence '1-2-3-4' followed by that
> same sequence again won't be prime, because:  ..."
>
> Are there any rules which describe prime numbers in such a way?
>

No, let N be any number then there is a prime containing a consecutive
subsequence of digits equal to N. One way to show this is as follows:
Let N = 921024..4. Now add the digit 1 to N to get M = 921024..41. Now
gcd(M,10^n) = 1. Choose n so that 10^n > M. Then k10^n + M will "contain"
M as least significant digits for all positive integers k. There is a
theorem due to Dirichlet which implies that there are infinitely many
primes of the form k10^n + M where n and M are fixed. See
http://mathworld.wolfram.com/DirichletsTheorem.html
for a statement of Dirichlet's Theorem.
• ... Here s an example constructed using Maple: Start with the number M. Just make sure its least significant digit is not 0, 2 or 5. ...
Message 3 of 4 , Aug 4, 2004
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On Wed, 4 Aug 2004, Edwin Clark wrote:

> On Wed, 4 Aug 2004, isbat1 wrote:
>
> > I'm interested in the idea that a number can be ruled out as a prime
> > number due to patterns occuring within the digits of that number.
> > For example, you might say, "a number containing five nines in a row
> > can't (or likely won't) be a prime number, for this reason: ...",
> > or "any number containing the sequence '1-2-3-4' followed by that
> > same sequence again won't be prime, because:  ..."
> >
> > Are there any rules which describe prime numbers in such a way?
> >
>
> No, let N be any number then there is a prime containing a consecutive
> subsequence of digits equal to N. One way to show this is as follows:
> Let N = 921024..4. Now add the digit 1 to N to get M = 921024..41. Now
> gcd(M,10^n) = 1. Choose n so that 10^n > M. Then k10^n + M will "contain"
> M as least significant digits for all positive integers k. There is a
> theorem due to Dirichlet which implies that there are infinitely many
> primes of the form k10^n + M where n and M are fixed. See
>   http://mathworld.wolfram.com/DirichletsTheorem.html
> for a statement of Dirichlet's Theorem.
>

Here's an example constructed using Maple:

Start with the number M. Just make sure its least significant digit is not
0, 2 or 5.

> M:=1111122222333330000004444444441:
> for i from 1 do if 10^i > M then n:=i; break; fi;od:
> print(M,n);
1111122222333330000004444444441, 31

Now we generate lots of primes of the form k*10^n + M:

> for k from 1 to 1000 do
> p:=k*10^n+M;
> if isprime(p) then print(k,p); fi;
> od:

24, 241111122222333330000004444444441

46, 461111122222333330000004444444441

48, 481111122222333330000004444444441

144, 1441111122222333330000004444444441

157, 1571111122222333330000004444444441

168, 1681111122222333330000004444444441

199, 1991111122222333330000004444444441

228, 2281111122222333330000004444444441

232, 2321111122222333330000004444444441

238, 2381111122222333330000004444444441

261, 2611111122222333330000004444444441

277, 2771111122222333330000004444444441

282, 2821111122222333330000004444444441

301, 3011111122222333330000004444444441

310, 3101111122222333330000004444444441

312, 3121111122222333330000004444444441

337, 3371111122222333330000004444444441

370, 3701111122222333330000004444444441

385, 3851111122222333330000004444444441

403, 4031111122222333330000004444444441

415, 4151111122222333330000004444444441

421, 4211111122222333330000004444444441

454, 4541111122222333330000004444444441

493, 4931111122222333330000004444444441

522, 5221111122222333330000004444444441

532, 5321111122222333330000004444444441

598, 5981111122222333330000004444444441

622, 6221111122222333330000004444444441

652, 6521111122222333330000004444444441

693, 6931111122222333330000004444444441

729, 7291111122222333330000004444444441

772, 7721111122222333330000004444444441

778, 7781111122222333330000004444444441

781, 7811111122222333330000004444444441

825, 8251111122222333330000004444444441

826, 8261111122222333330000004444444441

862, 8621111122222333330000004444444441

888, 8881111122222333330000004444444441

940, 9401111122222333330000004444444441

946, 9461111122222333330000004444444441

996, 9961111122222333330000004444444441

>
• Correction to my previous example. The example is okay, I just should have
Message 4 of 4 , Aug 4, 2004
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Correction to my previous example. The example is okay, I just should have
said:

> Start with the number M. Just make sure its least significant digit is not
> 0,2,4,5,6,or 8, so that gcd(M,10^n) = 1.
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