p*q = [5*7=35, 5*11=55, 7*11=77, 5*23= 115,

__7*23 = 161__, 5*47 = 235, 11*23 = 253, 5*59 = 295,

__7*47 = 329__, 7*59 = 413, 5*83 = 415, 11*47 = 517, 5*107 = 535, 7*83 = 581, 7*107 = 749, 5*167 = 835, 5*179 = 895,

__23*47 =1081__, 5*227 = 1135,

__7*167 = 1169,__7*179 = 1253, 5*263 = 1315, 7*227 = 1589,...]

phi(p*q) = [24, 40, 60, 88, 132, 184, 220, 232, 276, 348, 328, 460, 424, 492, 636, 664, 712,

__1012__, 904, 996, 1068, 1048, 1356,..]

order of 2 mod p*q = [12, 20, 30, 44,

__33__, 92, 110, 116,

__69,__174, 164, 230, 212, 246, 318, 332, 356,

__253__, 452,

__249__, 534, 524, 678]

**A few things popped up right away and led to the following definition:**

**Definition:**

**Tough primes**are primes q of the form 2p + 1 where p is a Sophie Germain prime such that q cannot also be represented as 8n + 7 for some n in Z.

In other words the set of tough primes is the set of safe primes without its intersection with the set of primes of the form 8n + 7 for some n in Z

The first few tough primes are 5, 11, 59, 83, 107, 179, 227, 347

The motivation behind defining tough primes is the following conjecture:

**Conjecture: The order of 2 mod p*q when p and q are tough primes is precisely phi(p*q)/2**

**Edit:**The order of 2 mod p*q for a tough prime p and a safe prime q that can also be represented as 8n+7 is also phi(p*q)/2.