Math moment

Yesterday, I noticed another thing that my wife in I have in common. And by that I mean, "have had in common since before we met." After all, it's not particularly remarkable to develop a number of things in common once you're married. But this new discovery adds to an already long list which includes (but is not limited to):

Both of us are firstborn.
Both born on a Sunday.
Have same number of letters in our first names.
Have (and had) same number of letters in our surnames.
Both have one sibling, a younger sister with 'Nicole' in her name.
Both fathers have 'Ray' in their names.
Our mothers' first names are Sheryl and Sharon.
Both have a Grandma Dorothy.
Both fathers were engineers, graduates of the same college.
Both mothers were nurses.

Hard to say how remarkable such stuff is. It is in our nature to take note of patterns and coincidences. So here's the new one I just noticed: Both my wife and I have social security numbers whose prime factorizations are three and one other number. Put another way, my SSN divided by three is a prime number, and so is my wife's SSN.

But this leads to the question: Is this really rare? Maybe lots of SSNs are this way. How are we to know, aside from undertaking a brute force operation sampling a great many numbers between our two SSNs? Interestingly, prime number theory tells us that there are shortcuts to these answers.

I'm going to have to dance around a little here to avoid publicly giving out too many hints about my SSN. Suffice it to say, the numbers for my wife and I start with the digits '39', and then differ. So let's re-state the question: "What is the spacing of numbers in the vicinity of 390 million where the numbers factor into two primes, one of which is three?" Well, this of course relates directly to the spacing of primes in the vicinity of 130 million. For example, the number 130,000,001 is prime. The next higher prime is 130,000,007. If we multiply both these numbers by three, we get two neighboring SSNs in the vicinity of 390 million that factor into two primes, one of which is three. In this case, the spacing between the SSNs is 18 (the difference between 390-00-0003 and 390-00-0021), or three times the difference between the two original primes. But is 18 a typical spacing?

Well, mathematicians in the 19th century figured out that the average spacing of primes in the neighborhood of a number N is very close to the natural logarithm of N, written ln(N). Furthermore, this formulation becomes more and more accurate the larger N is. This is a consequence, or perhaps an alternate way of stating, the Prime Number Theorem. Since I'm not a mathematician I'm probably not stating it well, but there it is. So using the Prime Number Theorem we can characterize the average spacing of primes in the vicinity of 130 million, and by extension the spacing of 3-times-prime numbers around 390 million.

From there, the answer falls out rather easily. The natural logarithm of 130 million is about 18.7. Multiply by 3 and you get 56. So on average, one out of every 56 SSNs starting with '39' factorize into three and a prime.

I'm going to stick to telling my wife she's one-in-a-million. One in 56 just isn't quite as flattering.