From: bobs@rsa.com Subject: Re: conjectures that are true for large n but are actually false Date: Wed, 27 Jan 1999 21:51:22 GMT Newsgroups: sci.math Keywords: Maximum number of primes in intervals (prime constellations) In article <78nnej$e60$1@nnrp1.dejanews.com>, > > >The cannonical example may be Merten's conjecture. > > > > I wouldn't call it canonical. It was mentioned at the start of the first > > issue of the _A M Monthly_ in 1987, but it is not a good example because > > it is heuristically false, and the heuristics suggest that it fails for > > such a large number. > > That is what you get if you invert an iterated logarithm. > > > > Douglas Zare > > > It's in fact easy to get similar conjectures, probably false, but true up to > some huge number. For instance, there is no known n greater than 4 such that > the interval n, n+100 contains 25 prime numbers, Unless I have counted wrong, the interval from 13 to 113 contains 25 primes... The interval [2,100] contains 25 primes. From 13 to 113 you lose 2,3,5,7,11 but pick up 101, 103, 107, 109, 113. [13,113] is the LAST such interval. Schinzel (I believe) was the first to prove (and I found a second purely numerically oriented proof while working on the convexity of pi(x)) that the best one can do * infinitely often* is to find 23 primes in an interval of length 100. > yet generalised "modulus" > conjectures give the existence of infinite such n, smallest around 10^100 Actually, one can show based on elementary congruence conditions that 25 is impossible i.o. I have done numerical work on the pi(x) conjecture. i.e. pi(a + b) < pi(a) + pi(b) Work of Hensley & Richards showed it is incompatible with the prime k-tuples conjecture and that if we designate b as the smaller of a,b, then if a counter example exists then b < 2000. My numerical work has shown b > 750 for a counter example. This problem is within computer reach. I don't have the CPU resources anymore to continue with it. -----------== Posted via Deja News, The Discussion Network ==---------- http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: conjectures that are true for large n but are actually false Date: 27 Jan 1999 22:01:01 GMT Newsgroups: sci.math wrote: >It's in fact easy to get similar conjectures, probably false, but true up to >some huge number. For instance, there is no known n greater than 4 such that >the interval n, n+100 contains 25 prime numbers, yet generalised "modulus" >conjectures give the existence of infinite such n, smallest around 10^100 Not so. I asked Maple to mark off all residues mod 210 which were not multiples of 2, 3, 5, or 7. Then I asked it to look at (circularly) consecutive strings of 101 entries and count how many entries were still unmarked. Indeed, there were nine such strings with 25 entries where primes could fall. I widened the scope to residues mod 2310, looking for strings with enough numbers which were at least not divisible by 2,3,5,7, or 11. Again there were strings with 25 such candidates, but only two of them (13..113 and the corresponding negatives mod 2310). However, the 25 candidates occupy all 13 different congruence classes mod 13, so in particular one of them will be divisible by 13. But of course the point is well taken: just as the twin-primes conjecture asserts the existence of pairs of primes in an arrangement not obviously excluded by divisibility mod 2, we can look modulo the primes 2 and 3 and expect infinitely many sets of primes of the form {n, n+2, n+6, n+8, n+12} (we must exclude n+14, then, since otherwise one of the six would be a multiple of 5). Or, going further, we expect sets of primes -- indeed, infinitely many of them -- of various other constellations of values. Actually finding them, however, is another matter (and proving there are infinitely many is beyond our skills at the moment.) dave (OK, I admit it; I went another step. There are seven +- pairs of strings of 101 residue classes modulo 30030 which have 24 entries not divisible by any of 2,3,5,7,11,and 13. Of these, most fall into 17 congruence classes modulo 17, and so cannot yield 24 primes, but the strings 6943 ... 7043 11563 ... 11663 each constitute only 16 classes modulo 17, and so might yield 24 integers prime to 17. Each set also falls into 16 (resp. 17) congruence classes modulo 19, and 18 (resp 19) congruence classes mod 23, so it is possible to find strings of 101 consecutive congruence classes mod 2*3*5*7*11*13*17*19*23 which contain 24 elements not divisible by any of these small primes. Since the 24 elements cannot exhaust the congruence classes of any larger primes either, we can therefore conjecture that for some n, all these values are prime. The smallest integer in the set will be congruent to one of 108457, 217153, 293257, 401953 modulo 510510. You can try looking at the strings of 101 starting with 510510*n+108457, say, but it would be about a one-in-a-million chance that all 24 terms would be prime to 19,23, ..., 97. But then you're looking at terms on the order of 10^12, say, and for a large N not divisible by a prime <100, that N is prime with probability about 8/log(N). Seems like the odds are against you until you've tried about 10^32 values of n . I guess I won't wait...)