perplexus.info :: Sequences : Prime, but not forever

Prime, but not forever (Posted on 2010-08-12)

Cosider the following sequence :
a(0) = p, where p is a prime
a(i+1) =2*a(i)+1) : e.g. 5, 11, 23, 47,...
Prove that there is no value of p creating a sequence of prime numbers only.

No Solution Yet Submitted by Ady TZIDON

Rating: 3.0000 (1 votes)

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proof

Comment 1 of 1

now we have

a(i) = (2^i) * p + (2^i) - 1

first I will prove it for p=2 and then prove it for p>2

if p=2 then we have

a(0)=2

a(5)=95=5*19

thus p=2 fails to create an infinite sequence of primes

now if p is a prime greater than 2 then we have

a(i)=2^i*p+2^i-1

by euler's totient theorem we have

2^(p-1) = 1 mod p

thus when i=p-1 then 2^i-1 is a multiple of p and thus we have

a(p-1) = 2^(p-1)*p + k*p for some integer k and thus

a(p-1) is a multiple of p and thus is not prime

thus all such sequences starting at a prime will fail to be prime at least at p-1 if not sooner with the exception of p=2 which fails at i=5

Edited on August 12, 2010, 2:27 pm

Posted by Daniel on 2010-08-12 13:42:36

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