Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).
Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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How do you solve b)?
Transcribed Image Text:Let ø(n) be Euler's phi function.
(a) Show that
1+2' + 2² + 2³+.
+ 26(15)–1 = 0 (mod 15)
...
but
1+3' + 32 + 33 +
+ 36(15)–1 0 (mod 15).
...
(b) Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then
1+a + a² + a³ + ….+a®(n)-1 = 0 (mod n).
...
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