Euler’s trinomial
f(x)= x^2 + x + 41 produces 40 distinct primes for the 40 consecutive
values of x i.e. x = 0 to 39.

While
investigating Euler’s trinomial for higher values of x, I noted the
following interesting observation:

For x>39, the
next best set of primes (which I term it as Prime clusters) occurs for
13 consecutive values of x i.e. for x = 219 to 231. On further
computations for x up to 10^10, it is observed that there is no prime
cluster which occurs for 13 or more consecutive values of x. However,
following 4 prime clusters occurs for 12 consecutive values of x:

(i) x = 32899350
to 32899361

(ii) x =
621213392 to 621213403

(iii) x =
1364016872 to 1364016883

(iv) x =
4541868613 to 4541868624

Q. Find next sets of prime clusters for
12 or more consecutive values of x in famous Euler’s trinomial.