Mathematics desk
< August 23	<< Jul \| August \| Sep >>	Current desk >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 24

Concatenated primes

Consider the sequence of numbers given by concatenating the first n integers in reverse order (1, 21, 321, 4321...). The first prime value in the sequence occurs when n = 82. I haven't found any more for n <= 500. Are there any more prime numbers in the sequence? Are there infinite primes in the sequence? 150.135.210.86 (talk) 17:24, 24 August 2015 (UTC)[reply]

I don't know - but the OEIS is a great resource for this kind of thing - see their page and refs on the sequence here [1]. SemanticMantis (talk) 17:52, 24 August 2015 (UTC)[reply]

Heuristically there should be an infinite number of primes in the sequence: We know that a(n) isn't divisible by 2 and 5, but apart from that it probably behaves like a random integer for primality testing purposes, so by the Prime number theorem the probability that a(n) is prime is approximately

{\frac {2}{1}}\cdot {\frac {5}{4}}\cdot {\frac {1}{\ln(a(n))}}

. Excluding the range you've already tested, the expected number of primes remaining in the sequence is

\sum _{n=501}^{\infty }2.5/\ln(a(n))

. Given that

a(n)<10^{n(\log _{10}(n)+1)}

and that

\sum _{n=2}^{\infty }{\frac {1}{n\ln(n)}}

diverges, the expected number of primes is infinite. To get an idea how far you'd have to search to have a 50% chance of finding the next example, you could try solving

\sum _{n=501}^{N}2.5/\ln(a(n))\geq 1

for N with a computer. Egnau (talk) 22:42, 25 August 2015 (UTC)[reply]

oeis:A176024 says the next term is for n = 37765, found by Eric W. Weisstein in 2010. It has 177719 digits and is only a probable prime so far. PrimeHunter (talk) 01:09, 26 August 2015 (UTC)[reply]

The corresponding sequence base 2 contains primes for n = 2, 3, 4, 7 at least. --JBL (talk) 23:02, 26 August 2015 (UTC)[reply]

Up to

n=1000

we have 2, 3, 4, 7, 11, 13, 25, 97, 110. Strangely, this sequence isn't in OEIS. -- Meni Rosenfeld (talk) 10:01, 28 August 2015 (UTC)[reply]