cubes congruent to 2^n (mod 215)

enzocreti · 2020-02-14, 15:19

1^3 is congruent to 2^n (mod 215) n=0

6^3 is congruent to 2^n (mod 215) n=0

7^3 is congruent to 2^n (mod 215) n=7

any other cubes congruent to 2^n (mod 215)?

Dylan14 · 2020-02-14, 16:56

Firstly, since n+215k is n mod 215 for any n and k in the integers, we know that (1+215k)^3 must be 1 mod 215 = 2^0 mod 215, and similarly for the others cases stated ((6+215k)^3 = 2^0 mod 215, and (7+215k)^3 = 2^7 mod 215).

So it suffices to check on the integers in the set [0, 214], then use the equivalence classes to quickly generate the rest.

2020-02-14, 15:19	#1
enzocreti Mar 2018 17×31 Posts	cubes congruent to 2^n (mod 215) 1^3 is congruent to 2^n (mod 215) n=0 6^3 is congruent to 2^n (mod 215) n=0 7^3 is congruent to 2^n (mod 215) n=7 any other cubes congruent to 2^n (mod 215)? Last fiddled with by enzocreti on 2020-02-14 at 15:23

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
sum of two positive cubes congruent to 1 mod Mersenne prime	enzocreti	enzocreti	2	2020-02-14 12:36
Fibonacci number as sum of cubes	jux	Miscellaneous Math	15	2015-08-30 06:21
Squares and Cubes:	mfgoode	Puzzles	24	2007-08-06 16:20
Prime cubes!	fivemack	Puzzles	4	2007-07-04 00:18
Counting Cubes	Numbers	Puzzles	6	2005-09-03 00:26

2020-02-14, 16:56	#2
Dylan14 "Dylan" Mar 2017 574₁₀ Posts	Firstly, since n+215k is n mod 215 for any n and k in the integers, we know that (1+215k)^3 must be 1 mod 215 = 2^0 mod 215, and similarly for the others cases stated ((6+215k)^3 = 2^0 mod 215, and (7+215k)^3 = 2^7 mod 215). So it suffices to check on the integers in the set [0, 214], then use the equivalence classes to quickly generate the rest.