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 2×5×53 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

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2020-02-14, 16:56	#2
Dylan14 "Dylan" Mar 2017 1001000011₂ 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.