Three sequences that I submitted to the OEIS (Online Encyclopedia of Integer Sequences) have now been approved.

The first sequence is the sequence generated by powers of 2 mod n when n is the product of 2 distinct safe primes. The sequence number is A269453. Here are the first few terms of the sequence:

12, 20, 30, 44, 33, 92, 110, 116, 69, 174, 164, 230, 212, 246, 290, 318, 332, 356, 410, 253, 452, 249, 530, 534, 524, 638, 678, 692, 716, 830, 393, 902, 764, 890, 932, 956, 1038, 1166, 1130, 537, 1004, 573, 1334, 1124, 1310, 1172, 1398, 717, 753, 1436, 1730, 913, 1886, 1686, 1790

Writing out this sequence along with the sequence of the products of two distinct safe primes is what led me to another, more interesting and definitely more useful sequence: the sequence of tough primes.

Tough primes are a subset of safe primes, which have interesting implications for the key generation in many public key cryptosystems.

By definition, tough primes are safe primes not congruent to -1 mod 8. Here are the first few tough primes:

5, 11, 59, 83, 107, 179, 227, 347, 467, 563, 587, 1019, 1187, 1283, 1307, 1523, 1619, 1907, 2027, 2099, 2459, 2579, 2819, 2963, 3203, 3467, 3779, 3803, 3947, 4139, 4259, 4283, 4547, 4787, 5099, 5387, 5483, 5507, 5939, 6659, 6779, 6827, 6899, 7187, 7523

Update: The sequence of tough primes is now sequence A269454 in the Online Encyclopedia of Integer Sequences. I unofficially name this sequence Yona after my great grandmother who taught me how to count.

## 25.3.16

### Sequencing The Unsequencable

Labels:
DH Problem
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Discrete Mathematics
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Mathematics
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Number Theory
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RSA Problem