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Old 2010-08-02, 13:43   #1
allasc
 
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Estimate please the test for simplicity of number

http://oeis.org/classic/A175625

a(i)=2*i+7; ((2*i+7) mod 3)> 0; (a(i) does not share on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3))

The first compound number a(i)=536870911
It is found by the user "venco" (from http://dxdy.ru)
It is interesting to notice that i+3=268435455=2^28-1

The second compound number a(i)=46912496118443
It is found by the user "venco"
It is interesting to notice that i+3=268435455=(2^46-1)/3

Exceptions are defined if to prove that they share on numbers (2^x-1) or (2^x+1)

Excuse for my English
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Old 2010-08-03, 06:13   #2
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Quote:
Originally Posted by allasc View Post
Estimate please the test for simplicity of number

http://oeis.org/classic/A175625

a(i)=2*i+7; ((2*i+7) mod 3)> 0; (a(i) does not share on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3))

The first compound number a(i)=536870911
It is found by the user "venco"
It is interesting to notice that i+3=268435455=2^28-1

The second compound number a(i)=46912496118443
It is found by the user "venco"
It is interesting to notice that i+3=268435455=(2^46-1)/3

Exceptions are defined if to prove that they share on numbers (2^x-1) or (2^x+1)

Excuse for my English
sorry :)

i+3=23456248059221=(2^46-1)/3

Last fiddled with by wblipp on 2010-08-03 at 19:52
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Old 2010-08-03, 17:34   #3
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I don't understand the sequence. Can you explain how you get 11, 23, 31, ...?
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Old 2010-08-03, 19:31   #4
allasc
 
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Quote:
Originally Posted by CRGreathouse View Post
I don't understand the sequence. Can you explain how you get 11, 23, 31, ...?
For any whole (i) condition performance
(2*i+7) mod 3)> 0; (a(i) does not share on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3)

Then the number 2*k+7 will be simple
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Old 2010-08-03, 19:54   #5
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Quote:
Originally Posted by allasc View Post
For any whole (i) condition performance
(2*i+7) mod 3)> 0; (a(i) does not share on 3); 4^(i+3) == 1 (mod (2*i+7)); 4^(i+2) == 1 (mod (i+3)

Then the number 2*k+7 will be simple
I take it from your .ru email address that by "simple" you mean prime and by "compound" you mean composite?

"(2*i+7) mod 3)> 0", I guess, means that 2i + 7 is not divisible by 3.

My guess is that your sequence is odd numbers a = 2i+7 such that
  • 2i + 7 is not divisible by 3;
  • 4^{i+3}\equiv1\pmod{2i+7}
  • 4^{i+2}\equiv1\pmod{i+3}
I haven't checked this yet.
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Old 2010-08-03, 20:03   #6
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Quote:
Originally Posted by CRGreathouse View Post
My guess is that your sequence is odd numbers a = 2i+7 such that
  • 2i + 7 is not divisible by 3;
  • 4^{i+3}\equiv1\pmod{2i+7}
  • 4^{i+2}\equiv1\pmod{i+3}
I haven't checked this yet.
It looks like the guess works out for the terms shown. I then take your statements to mean that an odd number a such that
  • a is not divisible by 3;
  • 2^{a-1}\equiv1\pmod{a}
  • 2^{a-3}\equiv1\pmod{(a-1)/2}
is some kind of probable-prime.
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Old 2010-08-03, 20:15   #7
allasc
 
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to 10^15 There are only two exceptions

Both exceptions divisible by expression 2^x-1 or 2^x+1 (If it was possible to prove it (What is the exception has such dividers).... It would be good)

The first compound number a(i)=536870911
It is found by the user "venco" (from http://dxdy.ru)
It is interesting to notice that i+3=268435455=2^28-1

The second compound number a(i)=46912496118443
It is found by the user "venco"
It is interesting to notice that i+3=23456248059221=(2^46-1)/3

PS Such coincidence happens to mine only in the Indian cinema

Last fiddled with by allasc on 2010-08-03 at 20:23
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Old 2010-08-03, 20:25   #8
allasc
 
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Excuse for my English :)
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Old 2010-08-03, 20:46   #9
CRGreathouse
 
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Quote:
Originally Posted by allasc View Post
to 10^15 There are only two exceptions
I trust you're just checking lists of 2-pseudoprimes?
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Old 2010-08-03, 21:22   #10
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Аскар, опишите по-русски, и я или кто-нибудь другой переведет, ok?
Есть много "ложных друзей переводчика", например "простое" число это вовсе не "simple", a "prime".
(We will translate from Russian to English.)
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Old 2010-08-03, 21:30   #11
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Batalov, would you comment on the simple/prime and compound/composite issue? This came up on the seqfans list recently (with another native Russian speaker) and I was reminded of it here. I asked a Ukrainian colleague about it, but she doesn't know the math terms in Russian...

Last fiddled with by CRGreathouse on 2010-08-03 at 21:30
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