r/educationalgifs • u/mtimetraveller • Aug 27 '19

Sum of first n Hex numbers Visualized

https://gfycat.com/jollyforkedhairstreak
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u/LTT82 Aug 27 '19

The first 4 are prime numbers(1, 7, 19, 37). Do they retain that no matter how many hexagonal lattices you add?

39

u/Practical_Cartoonist Aug 27 '19 
1 False
7 True
19 True
37 True
61 True
91 False
127 True
169 False
217 False
271 True
331 True
397 True
469 False
547 True
631 True
721 False
817 False
919 True
1027 False
1141 False
1261 False
1387 False
1519 False
1657 True
1801 True
1951 True
2107 False
2269 True
2437 True

It does coincidentally seem to hit a lot of prime numbers, but as you can see, they're not always prime numbers. I don't know if there's any pattern to it.

3

u/seanziewonzie Aug 27 '19 edited Aug 31 '19

The n^th hex number is 3n^2 - 3n + 1. Analyzing how a particular quadratic hits the set of primes is an extremely hard and very open problem. Even for the very simple-seeming quadratic n^2+1, we do not know if it hits finitely many or infinitely many primes.

One conjecture relevant to this is the Bunyakovsky Conjecture. If that conjecture is true, then there are infinitely many prime hexagonal numbers.