Each prime number can be written as a
non-trivial product of two integers, that is, so that neither factor is 1 - I
had to notice that.

I admit it was a special
experience. At first hearing, this is truly an apocalypse in classical number
theory.

The shock is resolved by a
small detail of the situation: the two integers whose product gives the prime
are complex integers:

(4+5i)(4-5i) = 41

I found many primes among
similar, i.e. (k + 5i) (k-5i) numbers, their number being infinite (but this
should remain a hypothesis for the time being). These:

29, 41, 61, 89, 281, 349, 509, 601, 701, 809,
1049, 1181, 1321, 1789, 2141, 2729, 3389, 4649, 5209, 5501, 5801, 8861, 9241,
9629, 10429, 11261, 11689, 12569, 15401, 15901, 17449, 17981, 18521, 19069,
21341, 21929, 23741, 24989, 26921, 27581, 33149, 39229, 40829, 41641, 42461,
45821, 46681, 52009, 53849, 55721, 59561, 68669, 71849, 79549, 80681, 86461, 87641,
91229, 94889, 97369, 98621, 99881, 101149, 107609, 111581, 112921, 114269,
116989, 118361, 126761, 128189, 133981, 135449, 139901, 145949, 147481, 149021,
153689, 156841, 158429, 169769, 173081, 174749, 179801, 181501, 186649, 190121,
195389, 197161, 198941, 204329, 209789, 226601, 228509, 234281, 252029, 254041,
264221, 266281…

Similar sequences can be found
if any other number is entered in place of 5 in the above formula.
Interestingly, in the first four cases we all get a series that is known in
OEIS - but not because of the property indicated here, but by referring to the
expanded form of the related formula:

5, 13,
17, 29, 37, 41, 53, 61 - https://oeis.org/A002144

2, 5,
17, 37, 101, 197, 257 - https://oeis.org/A002496

5, 13, 29, 53, 173, 229 - https://oeis.org/A005473

13, 73,
109, 409, 1033 - https://oeis.org/A138353

17, 41,
97, 137, 241, 457 - https://oeis.org/A243451

After the above observation, a
number of exciting questions arose in me. I have already found the answer to
most of these. I will share these in a subsequent post.

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