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A082369 Numbers congruent to 13 mod 30. 5
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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

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Last modified June 23 02:44 EDT 2017. Contains 288633 sequences.