Prime
Moments

My daughter, Lyra, and I were talking about birthdays and prime numbers. We realized we’d both be prime at the same time, and then it hit me that my wife Sarah, Lyra, and I, for a window of time would all have prime ages. Lyra at 11, Sarah at 41, me at 43. Three primes all at once.

That is a prime moment.

I dug into it. Not only do we have a prime moment, we have four of them. The same pattern of ages repeats across our lifetimes: (11, 41, 43), then (29, 59, 61), then (41, 71, 73), and finally (71, 101, 103). The shape is the same every time. I jokingly started calling these Toups Primes.

That puzzled me. Why does this repeat?

You might know twin primes. Pairs of primes exactly 2 apart, like 11 and 13, or 29 and 31. There are infinitely many primes. Mathematicians conjecture there are also infinitely many twin primes (and by extension, infinitely many Toups Primes).

Twin primes are a simple form of a prime constellation: a pattern of offsets that produces all-prime tuples. Twins have the shape [0, 2]. Pick a prime p. If p+2 is also prime, you have a twin. My family’s constellation is [0, 30, 32]. Pick a prime p. If p+30 and p+32 are also prime, you have a Toups triple. These shapes repeat because the gaps define a shape, and that shape can land on different starting primes.

A prime constellation is a pattern of age gaps that produces all-prime tuples. Not every group has one. A prime moment is when real people with real birthdays pass through that pattern on the calendar. Some constellations produce a single moment, but a large number produce multiple moments.

Footnote on 2: the only even prime. Any constellation that includes it can occur at most once and never repeats. The finder uses only odd primes for the recurring patterns. That’s where the interesting structure lives.