# Primes, Twin Primes, and My Mom’s Bday

Today is my mom’s birthday and something very special happened… We’re 5 in our family, my brother, my sister, me, my mom and my dad, and all our ages are prime numbers: 29, 31, 37, 61, 67!

I thought this was very cool, and decided to explore more.

Just to recall, a prime number is a natural number greater than 1, that has only 1 and itself as a divisor. Twin primes are pairs of prime numbers of the form (p, p+2), for example (3, 5), (5, 7), (11, 13). And then we’ll generalize, but let’s start talking about birthdays.

The first 25 prime numbers (all the primes less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Thus, a person leaving 100 years will celebrate 25 prime birthdays.

Prime numbers are infinite, and so if a person could live infinitely many years, she’d celebrate infinitely many prime birthdays.

There are 8 twin prime birthdays below 100 years old:
(3, 5)
(5, 7)
(11, 13)
(17, 19)
(29, 31) — my brother and sister
(41, 43)
(59, 61)
(71, 73)
And so my sibling will have 3 more twin prime birthdays to celebrate, at
(41, 43), (59, 61), and (71, 73).

Interestingly enough, if my brother and sister were to live infinitely many years… well, we actually don’t know how many twin prime birthdays they would celebrate… according to the twin primes conjecture, there are infinitely many twin primes, but this hasn’t been proved yet.

If we look at my brother and me, instead of twin primes of the form (p, p+2) we should look at pairs of primes of the form (p, p+8). Here are the 9 pairs smaller than 100:
(3, 11)
(5, 13)
(11, 19)
(23, 31)
(29, 37) — us
(53, 61)
(59, 67)
(71, 79)
(89, 97)
In analogy to the twin primes, we don’t know if there are infinitely many of these pairs or not… actually we don’t know any integer n such that there are infinitely many prime pairs of the form (p,p+n). Surprisingly, however, we know that there exists a n smaller than 246, such that there are infinitely many prime pairs…