Problem

Test all odd numbers in the range from 233 to 241 for primality using the Miller-Rabin test with base 2.

Answer:

test  n=233

• 233-1=2^3 * 29, thus k=3, q=29
• a^q mod n=2^29 mod 233=1
• test returns"inconclusive"(probably prime)

test n=235

• 235-1=2^1 * 117 ,thus k=1 q=117
• a^q mod n=2^117 mod 235=222
• 222!=1 and 222!=235-1
• test returns "composite"

test n=237

• 237-1=2^2 * 59, thus k=2, q=59
• a^q mod n=2^59 mod 237=167!=1
• 167!=237-1
• 167^2 mod 237=160!=237-1
• test returns "composite"

test n=239

• 239-1=2^1 * 119
• 2^119 mod 239=1
• test returns "inconclusive"(probably prime)

test n=241

• 241-1=2^4 *15
• 2^15 mod 241=233
• 233^2 mod 241=64
• 64^2 mod 241=240=241-1
• test returns "inconclusive"(probably prime)

問題思考

為什麼237測試兩遍就停了？為什麼不繼續往下平方測試

看K的值就知道了