Is this "Fibonacci" fact known?
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From: Achava Nakhash, the Loving Snake on 9 Jun 2010 16:49 Hi Group, Let p be a prime number and L(p) the p'th Lucas number. It is well- known and quite easily proved that L(p) = 1 (mod p) I believe that I have shown that any prime q dividing L(p), then q = 1 (mod p) except for the case when p = 3. An immediate consequence of this result is that given any prime p, there are infinitely many primes congruent to 1 mod p and the smallest one is no larger than L(p) <= 1 + (phi*p), where phi is the positive root of x^2 - x - 1 = 0. Is the result about prime divisors of L(p) known? Thanks, Achava
From: Gerry Myerson on 9 Jun 2010 19:12 In article <c3a9ab5d-1c1d-41b1-a75b-c78379f13fb3(a)34g2000prs.googlegroups.com>, "Achava Nakhash, the Loving Snake" <achava(a)hotmail.com> wrote: > Hi Group, > > Let p be a prime number and L(p) the p'th Lucas number. It is well- > known and quite easily proved that > > L(p) = 1 (mod p) > > I believe that I have shown that any prime q dividing L(p), then q = 1 > (mod p) except for the case when > > p = 3. An immediate consequence of this result is that given any > prime p, there are infinitely many > > primes congruent to 1 mod p and the smallest one is no larger than > L(p) <= 1 + (phi*p), where > > phi is the positive root of x^2 - x - 1 = 0. Wouldn't L(p) be more like phi-to-the-p, rather than phi-times-p? -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Achava Nakhash, the Loving Snake on 9 Jun 2010 20:41
On Jun 9, 4:12 pm, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email> wrote: > In article > <c3a9ab5d-1c1d-41b1-a75b-c78379f13...(a)34g2000prs.googlegroups.com>, > "Achava Nakhash, the Loving Snake" <ach...(a)hotmail.com> wrote: > > > > > > > Hi Group, > > > Let p be a prime number and L(p) the p'th Lucas number. It is well- > > known and quite easily proved that > > > L(p) = 1 (mod p) > > > I believe that I have shown that any prime q dividing L(p), then q = 1 > > (mod p) except for the case when > > > p = 3. An immediate consequence of this result is that given any > > prime p, there are infinitely many > > > primes congruent to 1 mod p and the smallest one is no larger than > > L(p) <= 1 + (phi*p), where > > > phi is the positive root of x^2 - x - 1 = 0. > > Wouldn't L(p) be more like phi-to-the-p, rather than phi-times-p? > > -- > Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)- Hide quoted text - > Of course. So the smallest prime congruent to 1 mod p is less than 1 + (phi^p) My fingers never seem to press the right key at the critical moment. Especially with the tiny keyboard on this laptop. Regards, Achava |