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• OOPs! I left out a in my definition of Legendre s conjecture. Legendre s conjecture is of course for n=2 and m = 1. My apologies Bill
Message 1 of 5 , Dec 9 9:50 AM
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OOPs! I left out a ">" in my definition of Legendre's conjecture.
Legendre's conjecture is of course for n=2 and m >= 1.
My apologies Bill
• ... Intuitively Legendre s conjecture would appear to be true: the rate of growth of the gap between squares looks like it would far exceed the rate of growth
Message 1 of 5 , Dec 12 2:13 AM
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--- In primenumbers@yahoogroups.com, "Bill" <bill2math@...> wrote:
>

> Legendre's conjecture claims there is at least one prime between the squares of consecutive integers, and is Capelle's case with n=2 and m = 1.
>

Intuitively Legendre's conjecture would appear to be true: the
rate of growth of the gap between squares looks like it would
far exceed the rate of growth in the average gap between primes.
However the pattern of these gaps can be very uneven and I believe that
it HAS been proven that the gap between primes can be arbitrarily
large. This does not disprove Legendre's conjecture - probably
the only way to do that would be to find a counterexample. Good
with that project!

a.
• ... The proof is trivial. For prime p, all integers n s.t. p!+1
Message 1 of 5 , Dec 12 10:31 AM
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On Sun, 2010-12-12 at 10:13 +0000, Aldrich wrote:
> However the pattern of these gaps can be very uneven and I believe
> that
> it HAS been proven that the gap between primes can be arbitrarily
> large. This does not disprove Legendre's conjecture - probably

The proof is trivial. For prime p, all integers n s.t.
p!+1 < n <= p!+p are divisible by at least one prime <= p and p can be
taken arbitrarily large.

Paul
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