Problems & Puzzles: Collection 20th

Coll.20th-014. Prime cluster in Euler's trinomial

On April 14, 2018, Shyam Sunder Gupta, wrote:

Eulerís 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 Eulerís 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 Eulerís trinomial.


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