Problems & Puzzles:
Collection 20th
Coll.20th-014.
Prime cluster in Euler's trinomial
On April 14, 2018,
Shyam Sunder Gupta,
wrote:
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.
On June 21, 2021 Giorgos
Kalogeropoulos wrote:
After x = 219 to 231, next prime cluster for 13 consecutive
values occurs for x=25529534712 to 25529534724
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