Conjecture : If $\ q\ $ is an odd prime, there is always a prime $\ p<q\ $ such that $\ p$#$+q\ $ is prime.
$p$# $= 2\cdot 3\cdot 5\cdots p\ $ is the primorial
This conjecture holds upto at least $\ 2\cdot 10^6\ $.
If we go upto $\ p=3200\ $ , the following $\ 41\ $ primes $\ q\le 10^8\ $ remain unsolved
[2173663, 9069217, 12759883, 15507827, 19582841, 19910833, 22686179, 25852909, 2
6277593, 28131179, 31054651, 34916879, 37117099, 38415271, 39765263, 40138019, 4
1140279, 43881083, 48265883, 52680623, 53129933, 57269941, 58942783, 59830109, 6
1308073, 61390057, 61392613, 61533467, 64493843, 68988053, 71345369, 73913599, 7
7045201, 79594259, 80342609, 82071553, 83446267, 85440581, 86935411, 93010331, 9
4369643]
For a large prime $q$, heuristically there should be a prime of the form $\ p$#$+q\ $ because such a number has no prime factor upto $p$, so the chance should be good that it is prime. Is there any hope to prove this conjecture ?
Update : The number $\ 45979$#$+25852909\ $ is probable prime.