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June 17th, 2009, 09:51 AM   #1
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Strange observation with primes

I was just multiplying consecutive prime numbers and i observed this .
Let p ,q ,r,s be any four consecutive prime numbers ( in ascending order).
Then the following results are observed :
1) Do p*q + r*s = N.
Check if N is divisible by 5 or not this is trivial if N is divisible by 5 u will have to choose a new set of four consecutive numbers or in the very beginning itself p,q,r,s is choose in a such a way that n is not divisible by 5.
Then divide N completely by 2 . i.e keep dividing by 2 till the numerator(Nr) becomes an odd integer
Then either the Nr is prime or the digits of the Nr occur in some prime
Example :
2*3 + 5*7 = 41
41*43 + 47*53 = 4254 = ( 4254/2 = 2127) this sequence occurs in a large number of prime numbers to list a few
i.e 21277 this prime.
I also observed something else also i am not very sure of this result though . Say Nr has k digits then there always exists a prime of k+1 digits which have the k digits of Nr in the same manner .
this is trivial in the above example .
I just will add few more examples for the sake of fun
21851 * 21859 + 21863 * 21871 = 955806682 ( Nr = 477903341) 5477903341 is a prime number!

I just don't have any idea at this moment on how to prove this or even if this is true for all primes.
Anyhelp would be nice . Thank you
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June 17th, 2009, 11:48 AM   #2
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Re: Strange observation with primes

Quote:
Originally Posted by kaushiks.nitt
or the digits of the Nr occur in some prime
I'm pretty sure this condition makes the whole thing trivial, since I expect you can find any digit sequence in a some (large enough) prime.

Partial Pari code:
Code:
test(p)=my(q=nextprime(p+1),r=nextprime(q+1),s=nextprime(r+1));test1(p*q+r*s)
test1(n)=if(n%5==0,0,test2(n>>valuation(n,2)))
test2(nr)=if(isprime(nr),0,nr)
forprime(p=2,1e3,if(test(p),print(p"\t"test(p))))
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June 17th, 2009, 06:41 PM   #3
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Re: Strange observation with primes

Quote:
I'm pretty sure this condition makes the whole thing trivial, since I expect you can find any digit sequence in a some (large enough) prime.
.
I know this is trivial but i mentioned that k digit sequence occurs in the prime having k+1 digit itself.
This is was interesting .
I am not interested in large enough prime though as this is trivial .
Please have a look at my previous post and check for the validity because i don't know even if it is true in all cases and hence no proof for might be a good conjecture ( or observation)
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June 17th, 2009, 07:16 PM   #4
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Re: Strange observation with primes

Quote:
Originally Posted by kaushiks.nitt
I know this is trivial but i mentioned that k digit sequence occurs in the prime having k+1 digit itself.
Ah. In that case the first failure is at p = 223, which yields 51989. The list continues 421, 683, 757, 773, 941, 1231, 1483, 1499, 1579, 1867, 2269, 2297, 2351, 2377, 2467, 2473, 2677, 2837, 2903, 2909, 2939, 3109, 3121, 3271, 3467, 3491, 3623, 3677, 4093, 4219, 4297, 4357, 4463, 4567, 4603, 4729, 4831, 5021, 5051, 5227, 5419, 5521, 5527, 5557, 5569, 5623, 5743, 5807, 5821, 5851, 5867, 5927, 6073, 6323, 6379, 6491, 6661, 6737, 6833, 6997, 7001, 7121, 7193, 7229, 7237, 7481, 7523, 7561, 7573, 7687, 7717, 7759, 7867, 7877, 7879, 7907, 7993, 8069, 8081, 8209, 8263, 8287, 8581, 8647, 8663, 8779, 9007, 9049, 9103, 9209, 9343, 9463, 9521, 9547, 9643, 9679, 9811, 9851, ....

Code:
test(p)=my(q=nextprime(p+1),r=nextprime(q+1),s=nextprime(r+1));test1(p*q+r*s)
test1(n)=if(n%5==0,0,test2(n>>valuation(n,2)))
test2(nr)=if(isprime(nr),0,if(itprime(nr),0,nr))
itprime(n)=my(b=10^#Str(n));for(i=1,9,if(isprime(n+b*i)||isprime(n*10+i),return(i)));0
forprime(p=2,1e4,if(test(p),print(p"\t"test(p))))
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June 17th, 2009, 10:07 PM   #5
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Re: Strange observation with primes

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by kaushiks.nitt
I know this is trivial but i mentioned that k digit sequence occurs in the prime having k+1 digit itself.
Ah. In that case the first failure is at p = 223, which yields 51989. The list continues 421, 683, 757, 773, 941, 1231, 1483, 1499, 1579, 1867, 2269, 2297, 2351, 2377, 2467, 2473, 2677, 2837, 2903, 2909, 2939, 3109, 3121, 3271, 3467, 3491, 3623, 3677, 4093, 4219, 4297, 4357, 4463, 4567, 4603, 4729, 4831, 5021, 5051, 5227, 5419, 5521, 5527, 5557, 5569, 5623, 5743, 5807, 5821, 5851, 5867, 5927, 6073, 6323, 6379, 6491, 6661, 6737, 6833, 6997, 7001, 7121, 7193, 7229, 7237, 7481, 7523, 7561, 7573, 7687, 7717, 7759, 7867, 7877, 7879, 7907, 7993, 8069, 8081, 8209, 8263, 8287, 8581, 8647, 8663, 8779, 9007, 9049, 9103, 9209, 9343, 9463, 9521, 9547, 9643, 9679, 9811, 9851, ....

Code:
test(p)=my(q=nextprime(p+1),r=nextprime(q+1),s=nextprime(r+1));test1(p*q+r*s)
test1(n)=if(n%5==0,0,test2(n>>valuation(n,2)))
test2(nr)=if(isprime(nr),0,if(itprime(nr),0,nr))
itprime(n)=my(b=10^#Str(n));for(i=1,9,if(isprime(n+b*i)||isprime(n*10+i),return(i)));0
forprime(p=2,1e4,if(test(p),print(p"\t"test(p))))
Yeah but those are big numbers anyways.
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June 18th, 2009, 02:19 AM   #6
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Re: Strange observation with primes

Quote:
Ah. In that case the first failure is at p = 223, which yields 51989.
.
It doesn't fails at 223.
It is 211*233 + 227 *229 = 99036 = 24579 and 224579 is prime
I don't know how it failed for you . please check your code.

You mentioned it also fails for 421.
It doesn't 419 *421 + 431 *433 = 363022 = 181511 and 5181511 is prime
For 683 is also mine is right.
It is 677*683 + 691*701 = 946782 = 473391 and 1473391 is prime

I just couldn't code it and check if all your counter examples are wrong . But these are wrong for sure so please have a look at your code . I randomly choose some 1000 odd primes and checked .
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June 18th, 2009, 02:34 AM   #7
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Re: Strange observation with primes

Quote:
3109, 3121,
.
For these numbers it is bound to fail because the odd number is divisible by 5 .
I guess i mentioned if the even number which is obtained using the products is divisible by 5 then the test fails or in other words one shouldn't choose such numbers .
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June 18th, 2009, 04:57 AM   #8
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Re: Strange observation with primes

Quote:
Originally Posted by kaushiks.nitt
I don't know how it failed for you . please check your code.
That would be because you didn't specify your requirements well.

I'll let you modify the code to find a new list of failures.
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June 18th, 2009, 07:58 AM   #9
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Re: Strange observation with primes

Quote:
That would be because you didn't specify your requirements well.
.
I am sorry for not specifying the requirements . Anyway thanks .
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June 18th, 2009, 10:39 AM   #10
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Re: Strange observation with primes

I also made another observation all the odd numbers that appear when we apply collatz conjecture on a prime are themselves prime or semi prime . With the exception being if the odd number has 5 as one of it's factor then it might have two other factors than this .
Also something that is trivial is collatz conjecture need not be proved for all numbers it is enough if we prove it for all prime numbers from which we can prove that it is true for all numbers .
As i am not very sure about the first statement i made i would say it for sure that other than even power's of two and 5 one will definitely encounter at least one prime number when the collatz function is applied time and again still we reach 5 or 2.
This is basically the second statement.
As i don't have a powerful software and a computer i just couldn't test the result for all primes .
Thus it would be nice if someone could just check and tell the first observation of course the second is always true .
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