Primes Containing but Not Ending in 123456789Date: 02/26/2003 at 20:35:12 From: Marie Subject: Prime numbers Are there infinitely many primes that contain but do not end in the block of digits 123456789 ? I could do the problem if it asked me to show infinitely many primes that end in this block, because I would use Dirichlet's theorem. But I am unsure of how to represent the block in any position. Date: 02/27/2003 at 02:48:42 From: Doctor Jacques Subject: Re: Prime numbers Hi Marie, I understand that you want to use Dirichlet's theorem with your progression: 10^11 k + 1234567891 This progression contains infinitely many primes, and all these primes contain the string 123456789. So, you _did_ prove the theorem, and, with an obvious generalization, you could prove even more: There are infinitely many primes that contain the string 123456789 in any pre-assigned position. This is also true for other strings, except that you cannot have infinitely many primes ending in 5 or an even number (Dirichlet's theorem would not apply in this case). You said you could solve the problem with the string in the units position. Let us try to prove the theorem in the same way with the string 3 positions to the left (i.e. with the 9 in the thousands position). We want to show that there are infinitely many primes in the form: ...xxxxx123456789yyy As there are no constraints on the xxx and yyy, we are free to choose yyy - this will prove more than what is required. Pick for example, yyy = 537. Our problem is now reduced to proving that there are infinitely many primes ending in: 123456789537 and Dirichlet's theorem can be used in an obvious way, with the progression: n*(10^12) + 123456789537 All those (inifinitely many) primes contain 1234565789 in a position other than the last, so they witness the exactitude of the theorem. You do not need to have a general formula for the string in any position - you can pick any string yyy (that does not end in 5 or an event number), and append it to the right of the given string. Does this make sense? Write back if you want to discuss this further. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ |
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