A082369 Numbers congruent to 13 mod 30.

13, 43, 73, 103, 133, 163, 193, 223, 253, 283, 313, 343, 373, 403, 433, 463, 493, 523, 553, 583, 613, 643, 673, 703, 733, 763, 793, 823, 853, 883, 913, 943, 973, 1003, 1033, 1063, 1093, 1123, 1153, 1183, 1213, 1243, 1273, 1303, 1333, 1363, 1393, 1423, 1453

Offset

1,1

Comments

Solutions to 19^x + 23^x == 29 mod 31.

The form of the these numbers is obviously 30X + 13. 3^x + 5^x == 7 mod 11 and 17^x + 19^x == 23 mod 29 have no solutions. In fact, 3^x + 5^x == m mod 11 is only solvable for m < 11 = 1, 2, 8, 9. Similarly, 17^x + 19^x == m mod 29 is not solvable for m < 29 = 6, 11, 13, 14, 15, 16, 18, 23. I can't even prove 3^x + 5^x-7 <> 11k for all integers x, k. Anyone have a general proof of these statements say, a^x + -b^x == m mod k true or false for certain a, b, m, k, x combinations?

a^x + b^x == m (mod k) is periodic mod phi(k), so it suffices to check x = 1, 2, ..., phi(k). - Charles R Greathouse IV, Nov 19 2013

Links

Michael G. Kaarhus, Table of n, a(n) for n = 1..10000

Tanya Khovanova, Recursive Sequences

Index entries for linear recurrences with constant coefficients, signature (2,-1).

Formula

a(n) = 30n + 13.

G.f.: x*(13+17*x)/(1-x)^2. - Colin Barker, Jan 11 2012

Mathematica

Range[13, 7000, 30] (* Vladimir Joseph Stephan Orlovsky, Jul 13 2011 *)
LinearRecurrence[{2, -1}, {13, 43}, 50] (* Harvey P. Dale, Mar 02 2023 *)

Prog

(PARI) anpbn(n)= for(x=1, n, if((19^x+23^x-29)%31==0, print1(x" "))) \\ solutions to 19^x+23^x == 29 mod 31
(Maxima) f(a):= mod((19^a + 23^a), 31)$ a:-1$ for n:1 thru 3000 step 0 do(a:a+1, if f(a)=29 then (if mod(a, 30)=13 then (print(n, " ", a), n:n+1) else (print("Exception at ", a, ", ", f(a)), n:3001))) /* f(a)==29 only when a is cong. to 13 (mod 30). No exceptions thru a=89983, n=3000. Michael G. Kaarhus, Nov 18 2013 */$

Crossrefs

Sequence in context: A043921 A138686 A264900 * A132233 A282322 A031382

Adjacent sequences: A082366 A082367 A082368 * A082370 A082371 A082372

Keyword

nonn,easy

Author

Cino Hilliard, May 11 2003

Extensions

Simpler name from Charles R Greathouse IV, Nov 19 2013

Status

approved