12 doesn't miss 4; 2584 % 12 == 4. On Fri, Oct 9, 2020 at 7:43 AM Allan Wechsler <acwacw@gmail.com> wrote:
Oh, gosh. I stopped enumerating just shy of mod 8, and missed these.
I now see the point of this question. Mod 8 misses 4 and 6; mod 11 misses 4, 6, 7, and 9.
Unexpectedly, the missed sets get repeated: 12 misses 4 and 6, like 8; and 13 misses 4, 6, 7, and 9, like 11. 16 misses 4, 6, 10, 12, 14.
So far, every modulus that misses any residue at all, misses 4 and 6. Does that remain true?
Does OEIS have a sequence that tells which modulus first misses n? I can't find it because I can't guess how it handles residues (like the Fibonacci numbers) that are never missed.
On Fri, Oct 9, 2020 at 9:15 AM James Propp <jamespropp@gmail.com> wrote:
No Fibonacci number is congruent to 4 or 6 mod 8.
See http://oeis.org/A066853 (the sequence that gives the number of "Fibonacci residue classes" mod m for m=1,2,3,...) and http://oeis.org/A079002 (the sequence of moduli such that the Fibonacci numbers fill all m possible residue classes).
Jim
On Fri, Oct 9, 2020 at 8:10 AM Allan Wechsler <acwacw@gmail.com> wrote:
Perhaps I have misunderstood how the quantifiers are scoped an this question. I recall being convinced that the Fibonacci numbers hit all residue classes, in which case every integer is "pseudo-Fibonacci". If this is not true, can you name a residue class that contains no Fibonacci numbers?
On Thu, Oct 8, 2020, 11:07 PM James Propp <jamespropp@gmail.com> wrote:
Does there exist a positive integer n that isn’t a Fibonacci number, such that for every modulus m there is a Fibonacci number congruent to n mod m?
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