Question

prove 10.8-10.9

LLLLLLLLL think the converse to Fermar's Little Theorem is true? 10.8 Theorem. Lern be a natural number greater than 1. Then 7 is prime if and only if a"- = 1 (mod n for all natural numbers a less than n. 10.9 Question. Does the previous theorem give a polynomial or exponen- rial time primalin test? Inventing polynomial time primality tests is quite a challenge. One way to salvage some good from Fermat's Little Theorem is to weaken our demand of certainty. What if instead we look for a probable prime rest, by which we mean a statement of the form n is very likely to be prime. where the blank would be filled in by some testable condition on n.

Answer 1

10.8. let n bea nature numbes grcccbe a thoin 1 of n is a poime numbek, then by peonment theorem ) - 1 c modo fox albatical numbeisa less than n. D-1 conve mod for all Data numbers a less than ni Claimi- nis a prime. Assume on contraty that n is not psime. Then n = nying for some Ne, n2 <n. By Given, ni-2.1 cmod.n).&n =1 cmodn). ☆ (n7"-4-1) = 0 cmodn) # C92-2) = 0 cmodn) - + cnth-21)(n.) - 1) = 0 cmodn) + m2?? n)-2- 62-22-42) = o cmodn) 5 (11-2--1911 3 0 cmody Co -1 -1 +1) çocmodn) 0 : 1 (modn) contradicton Hence n must be a prime F F ť