"henh...@gmail.com" <henhanna@gmail.com> writes:
> Show that
> where p is a prime number ,
> ( p ^ 4 + 14 ) is not prime.
>
>
> (sorry if this too easy)
>
> pls wait at least 1 week posting answer(s) or hints.

Or other people could wait at least 1 week reading this follow-up.

Then again, the solution to this problem is simpler than the question
itself - if you understand the question, you should be able to answer
it.

Anyway, here's my hint...

Except for primes divisible by 3, p^4+14 is non-prime for exactly
the same reason that p^4+2 and p^4+8 are also non-prime. The 3|p
case is trivial.

Anyway, let's flip parity and turn this into a race.

For which N is p^4+N prime most often, for p<1000?

There are 168 primes p in that range, and I have found an N that yields
84 primes (so a full 50% of candidates).

Spoiler:

? forprime(p=2,1000,print(p" "factor(p^4+6156712)))
2 [2, 3; 769591, 1]
3 Mat([6156793, 1])
5 Mat([6157337, 1])
7 Mat([6159113, 1])
11 Mat([6171353, 1])
13 Mat([6185273, 1])
17 Mat([6240233, 1])
19 Mat([6287033, 1])
23 Mat([6436553, 1])
29 Mat([6863993, 1])
31 Mat([7080233, 1])
37 Mat([8030873, 1])
41 Mat([8982473, 1])
43 [349, 1; 27437, 1]
47 Mat([11036393, 1])
53 Mat([14047193, 1])
59 Mat([18274073, 1])
61 Mat([20002553, 1])
67 Mat([26307833, 1])
71 Mat([31568393, 1])
73 Mat([34554953, 1])
79 [47, 1; 959719, 1]
83 [3877, 1; 13829, 1]
89 [397, 1; 173549, 1]
97 [127, 1; 193, 1; 3863, 1]
101 Mat([110217113, 1])
103 [151, 1; 839, 1; 937, 1]
107 [6299, 1; 21787, 1]
109 [47, 1; 1697, 1; 1847, 1]
113 Mat([169204073, 1])
127 Mat([266301353, 1])
131 [887, 1; 338959, 1]
137 [193, 1; 1857161, 1]
139 Mat([379457753, 1])
149 [227, 1; 2198419, 1]
151 Mat([526042313, 1])
157 [127, 1; 181, 1; 26699, 1]
163 [5279, 1; 134887, 1]
167 [2221, 1; 352973, 1]
173 [47, 1; 907, 1; 21157, 1]
179 Mat([1032782393, 1])
181 [2609, 1; 413737, 1]
191 Mat([1337020073, 1])
193 [12799, 1; 108887, 1]
197 Mat([1512295193, 1])
199 [151, 1; 10426463, 1]
211 [10567, 1; 188159, 1]
223 [827, 1; 1009, 1; 2971, 1]
227 [1301, 1; 2045653, 1]
229 [2963, 1; 930211, 1]
233 Mat([2953452233, 1])
239 Mat([3268965353, 1])
241 [39521, 1; 85513, 1]
251 Mat([3975282713, 1])
257 Mat([4368627113, 1])
263 [1789, 1; 2677757, 1]
269 Mat([5242271033, 1])
271 Mat([5399737193, 1])
277 [2399, 1; 2456647, 1]
281 Mat([6240996233, 1])
283 Mat([6420404633, 1])
293 Mat([7376207513, 1])
307 Mat([8889030713, 1])
311 Mat([9361108553, 1])
313 Mat([9604081673, 1])
317 [1951, 1; 5178983, 1]
331 [281, 1; 42739393, 1]
337 Mat([12904074473, 1])
347 [1579, 1; 9185867, 1]
349 [389, 1; 1009, 1; 37813, 1]
353 Mat([15533559593, 1])
359 [523, 1; 1871, 1; 16981, 1]
367 Mat([18147283433, 1])
373 Mat([19363035353, 1])
379 [2063, 1; 10004311, 1]
383 [1049, 1; 20518417, 1]
389 Mat([22904201753, 1])
397 Mat([24846753593, 1])
401 [947, 1; 27310579, 1]
409 Mat([27989089673, 1])
419 [36319, 1; 848807, 1]
421 [167, 1; 3209, 1; 58631, 1]
431 Mat([34513305833, 1])
433 Mat([35158281833, 1])
439 Mat([37147540553, 1])
443 Mat([38519826713, 1])
449 [181, 1; 673, 1; 333701, 1]
457 [389, 1; 1993, 1; 56269, 1]
461 [743, 1; 821, 1; 74051, 1]
463 Mat([45960224873, 1])
467 Mat([47568968633, 1])
479 Mat([52649329193, 1])
487 [911, 1; 2437, 1; 25339, 1]
491 Mat([58126205273, 1])
499 Mat([62007654713, 1])
503 [2837, 1; 22565989, 1]
509 Mat([67129121273, 1])
521 Mat([73686373193, 1])
523 [193, 1; 3889, 1; 99689, 1]
541 [809, 1; 105894097, 1]
547 [47741, 1; 1875373, 1]
557 [10709, 1; 8988757, 1]
563 [1049, 1; 2389, 1; 40093, 1]
569 Mat([104827341833, 1])
571 [79153, 1; 1343081, 1]
577 [251, 1; 441625003, 1]
587 Mat([118733952473, 1])
593 Mat([123663175913, 1])
599 [236993, 1; 543241, 1]
601 [1259, 1; 103631707, 1]
607 [6571, 1; 20660603, 1]
613 [349, 1; 9323, 1; 43399, 1]
617 Mat([144930271433, 1])
619 [3203, 1; 45837811, 1]
631 [74231, 1; 2135743, 1]
641 [3137, 1; 53818729, 1]
643 [47, 1; 12907, 1; 281797, 1]
647 [179, 1; 739, 1; 1324753, 1]
653 Mat([181830791993, 1])
659 Mat([188606143673, 1])
661 Mat([190906116953, 1])
673 [47, 1; 30911, 1; 141209, 1]
677 [14327, 1; 14662639, 1]
683 [281, 1; 774441793, 1]
691 Mat([227994262073, 1])
701 Mat([241481099513, 1])
709 [431, 1; 1289, 1; 454847, 1]
719 Mat([267254832233, 1])
727 [19687, 1; 14189519, 1]
733 [1291, 1; 223613963, 1]
739 [109451, 1; 2725003, 1]
743 Mat([304764255113, 1])
751 [10601, 1; 30006913, 1]
757 Mat([328391313113, 1])
761 [13679, 1; 24518407, 1]
769 Mat([349713989033, 1])
773 Mat([357047062553, 1])
787 Mat([383624114873, 1])
797 Mat([403496630393, 1])
809 [1279, 1; 3547, 1; 94421, 1]
811 [181, 1; 2390072213, 1]
821 [433, 1; 1049278121, 1]
823 Mat([458780730953, 1])
827 [6221, 1; 75191293, 1]
829 [1409, 1; 335206777, 1]
839 [13879, 1; 35702207, 1]
853 Mat([529421013593, 1])
857 Mat([539421490313, 1])
859 [127, 1; 4287200999, 1]
863 [877, 1; 3457, 1; 182957, 1]
877 Mat([591565575353, 1])
881 [181, 1; 997, 1; 3338369, 1]
883 [397, 1; 3761, 1; 407149, 1]
887 [185957, 1; 3328789, 1]
907 Mat([676757533913, 1])
911 [433, 1; 1590704441, 1]
919 [127, 1; 5616452279, 1]
929 [181, 1; 251, 1; 16395103, 1]
937 Mat([770835721673, 1])
941 Mat([784082758073, 1])
947 [349603, 1; 2300531, 1]
953 Mat([824849744393, 1])
967 [5503, 1; 158894711, 1]
971 [769, 1; 1155988697, 1]
977 Mat([911131768553, 1])
983 Mat([933720588233, 1])
991 [6703, 1; 7829, 1; 18379, 1]
997 Mat([988060048793, 1])

Rejecting all primes under 47 as possible factors, and all other primes
under 127, gives this N a very high prime density.

Phil
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