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JocK
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Primes in base 3
« on: Aug 14th, 2005, 5:19am » |
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In base 3, apart from the second prime, all primes end in 1 or 2:
2, 10, 12, 21, 102, 111, 122, 201, 212, 1002, 1011, 1101, 1112, 1121, ...
Can you prove the following:
There is a value of N such that when above sequence is truncated after N terms, there are more terms that end in "1" than there are that end in "2".
We can try to extend this to any base b:
Can you prove that there is a N such that the first N primes when written in arbitrary base b contain more terms ending in "1" than those ending in any other specific symbol?
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
xy - y = x5 - y4 - y3 = 20; x>0, y>0.
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Barukh
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Re: Primes in base 3
« Reply #1 on: Aug 14th, 2005, 7:22am » |
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on Aug 14th, 2005, 5:19am, JocK wrote:| There is a value of N such that when above sequence is truncated after N terms, there are more terms that end in "1" than there are that end in "2". |
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So, actually, what you are asking to prove is that there is a point where primes of the form 3n+1 dominate primes of the form 3n-1? 
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Eigenray
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Re: Primes in base 3
« Reply #2 on: Aug 14th, 2005, 1:34pm » |
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Very interesting! In 1853 Chebyshev noted that primes seemd to be congruent more often to 2 mod 3 (or to 3 mod 4) than to 1, even though they are asymptotically equally distributed. In fact, for a given base, there is a bias towards quadratic non-residues in general [Michael Rubinstein and Peter Sarnak. "Chebyshev’s Bias", Experimental Mathematics, Vol.3, 1994 (pp. 173-197).]
If q is such that (Z/qZ)* is cyclic, let piR(x,q) and piN(x,q) denote the number of primes <=x which are quadratic residues and nonresidues, respectively, mod q, and let
Pq = {x : piN(x,q) > piR(x,q) }.
Then, assuming GRH and GSH (the hypothesis that the imaginary parts of the roots on the critical line of certain L functions are linearly independent over Q), Rubinstein and Sarnak in 1994 showed that Pq always has logarithmic density strictly between 1/2 and 1, and that it approaches 1/2 as q -> infinity. For q=3,4 the densities are about 0.9990 and 0.9959, respectively.
In particular, there are infinitely many such N as in Jock's question. That fact was proved by Littlewood in 1914; the first such N was found by Bays and Hudson in 1978.
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