Question

(1) Is 1387 prime? (6 marks) (a) Show that 1387 is a base 2 pseudoprime. (b) Show that 1387 is actually composite by using Miller's test in base 2.

Please explain each step carefully with any theorems that you use. Cheers!

Answer 1

Fermat's Little theorem P is a prime number, Then for any intege a, the numbeY aP.a is an integey multiple of P. a a(modp) if a is not divisible by P. a 2-Mod p) 2 Peucdo prime if 286 IMod 1387 1387 is a base a 1387 1973 note we 386: 13x7x11 By fermat liHle theorem 18-1mod 1nso (a) Modig Apply FLT 0gain 23.2 19)4mod 73 would then tike to imply 28-mod 73 the chinese Remainder theorem Then by ' ১৪৫ IMod 138 dy the statement given 1337 is a base 2 psuedoprime

bA numbey is comp0site if it does not Miller's test for Some pass base HeYe, we use base 2 1386 693 - 2 -512 mod 1387, 13862 is not congruent to 1 oY -1 modi a8 thus 2 Theyefoye 1387 is composite. 2693 512 mold 1387 have Calculated using calcutatol 693- 3.37.1t