"Accidental Order" in Pi, eDate: 04/12/99 at 20:44:06 From: collins titus Ngeno Subject: Prime numbers! The occurence of the sequence 1234567890 in the decimal expansion for pi or e may be called "accidental order." a) Show that this sequence appears in the digital interior of infinitely many prime numbers. b) Does the sequence consisting of the first billion digits of pi appear in the digital interior of infinitely many prime numbers? Please help. Titus Date: 04/14/99 at 13:02:43 From: Doctor Nick Subject: Re: Prime numbers! Hi Titus, I have to assume you are studying number theory, and that you know of Dirichlet's result on primes in arithmetic progression. This states that if r and s are relatively prime, then the set of positive integers r, r+s, r+2*s, r+3*s, ... contains infinitely many primes. We can use this result to answer questions like your a) and b). I'll do a simpler problem, and you can work out yours. Let's show that there are infinitely many primes that have "12" in their digits. We notice that primes can't end in "12," but they can end in "121." So, consider the arithmetic progression with r=121, s=1000: 121, 1121, 2121, 3121, 4121, ..., 10121, 11121, ... . Since 121 and 1000 are relatively prime, this progression contains infinitely many primes; hence, there are infinitely many primes which contain "12" in their digits. Notice that this doesn't come anywhere near accounting for _all_ primes that have "12" in their digits, but that's not needed. Also notice that we could have used r=123, s=1000, or r=1201, s=10000, etc. Now, you just have to apply this method to the larger digital sequences of a) and b). Write back if you get stuck. Have fun, - Doctor Nick, The Math Forum http://mathforum.org/dr.math/ |
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