Xi'an's Og

A prime number challenge [or rather two!] as Le weekly Monde current mathematical puzzle:

When considering the first two integers, 1 and 2, their sum is 3, a prime number. For the first four integers, 1,2,3,4, it is again possible to sum them pairwise to obtain two prime numbers, eg 3 and 7. Up to which limit is this operation feasible? And how many primes below 30,000 can write as n^p+p^n?

The run of a brute force R simulation like

max(apply(apply(b<-replicate(1e6,(1:n)+sample(n)),2,is_prime)[,b[1,]>2],2,prod))

provides a solution for the first question until n=14 when it stops. A direct explanation is that the number of prime numbers grows too slowly for all sums to be prime. And the second question gets solved by direct enumeration using again the is_prime R function from the primes package:

[1] 1 1
[1] 1 2
[1] 1 4
[1] 2 3
[1] 3 4

This entry was posted on December 29, 2019 at 12:19 am and is filed under Books, Kids, R with tags brute-force solution, Le Monde, mathematical puzzle, prime number, primes, R. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.