Recently I’ve come across some posts on Twitter that have led me off in unexpected directions. In short, they have served as prompts for me to ask and pursue answers to questions, processes that have given me both enjoyment and insight into some mathematical ideas and truths. I thought I would share one here.
It was prompted by this image and associated questions, from Christopher Danielson (@Trianglemancsd).
I’ve increasingly come to appreciate the opportunities these “How many … ?” prompts afford me to look beyond my immediate perception and initial responses. (Simon Gregg, @Simon_Gregg, and Hana Murray, @MurrayH83, are two others whose similar posts I have enjoyed engaging with.) With a bit of effort and practice I begin to see new and different ways to conceptualise the numbers represented in these arrays. Here’s what I came up with for this one.
I then realised the four peaches to the left could also be arranged to form a second square. In other words, 29 can be written as the sum of two squares. Not only that, but 29 and the “sides” of the two squares, 2 and 5, are all prime numbers.
So, by thinking about a picture of some peaches, I had found a prime that can be written as the sum of the squares of two smaller primes.
I wondered if there were other primes that could be written in this way. By following up on this wondering, not only did I find some more, I discovered two conditions that must be met if two primes are to combine in this way to make a larger one. I’ve since come to think, light-heartedly, of what I discovered as my “peach-pair prime theorem”.
In searching for more examples, it occurred to me to systematically pair primes from an ascending list: first pair 2 with 3, then with 5, 7, … . Next I would pair 3 with 5, 7, 11, …, then 5 with 7, 11, 13, … and so on, squaring and adding both primes in each pair (is it just me, or are you thinking Pythagoras too?) and checking whether or not each result was prime.
My first three attempts were fruitful – pun intended! (These included 29, the not unfortunate event that initiated this whole enquiry):
22 + 32 = 13
22 + 52 = 29
22 + 72 = 53
The next pair, though, generated 125, easily-recognisable as a composite number!
22 + 112 = 125 = 53.
My early findings, then, seemed to suggest there would be lots of primes that could be generated in this way, although clearly not every pairing would work.
I moved on, adding the square of 3 to those of 5, 7 and 11 respectively ? These produced composite sums of 34, 58 and 130 respectively. Surely I would eventually find some primes if I persisted with this phase of my search?
It was at this point that I stood back to reflect on the evidence so far. In doing so I discovered the two constraints referred to earlier. Before reading on, you might want to see if you can find the first one for yourself.
The pairs I had found all involved 2. We know that 2 is the smallest prime as well as the only even one, which makes it the oddest of all primes! (Credit for that observation goes to Peter M. Higgins, in his excellent little OUP book “Numbers, A very Short Introduction”). I realised this was the key to understanding why there was no point continuing to search for pairs with 3 as the smallest prime (or 5, 7, 11, … for that matter). Since all these other primes are odd, their squares must also be odd. The sum of the squares of two odd primes will always be both even and greater than 2. In other words, it cannot be prime. I had discovered a theorem! I imagine it has been discovered before, probably many times, but it was a new idea for me. Most importantly, reasoning alone allows us to be certain it is true.
Theorem (first draft): The only primes that can be expressed as the sum of the squares of two smaller primes are those where one, and only one, of the smaller primes to be squared is 2.
Now, as we have seen, this pairing and squaring process involving 2 and another prime does not always generate a prime. In fact, while there are many known algorithmic approaches to generating primes, none work in all cases. In any event, I had already found a counter-example in 22 + 112 = 125. Rather than just acknowledging it, though, I wondered if there was a reason why this one was different from my earlier successes. Again this probing approach led me to discover the second constraint that would mean only pairs of primes with exactly the right credentials – my “peach-pairs” – could potentially combine to form a larger prime. Before reading on I invite you, once more, to see if you can discover it for yourself.
Another way of stating my draft theorem above is to say all primes that are the sum of the squares of two smaller primes are 4 more than the square of an odd prime. Now an odd prime must end with one of the digits 1, 3, 5, 7 or 9. The squares of all numbers ending in either 1 or 9 must, themselves, end in 1. Adding 4 to these squares will produce a sum ending in 5. Hence, the result will be a composite number. My theorem was now more precise.
A peach-pair prime theorem: The only primes that can be expressed as the sum of the squares of two smaller primes are those where the two smaller primes are 2 and a prime ending with either of the digits 3, 5 or 7.
Before concluding this article, I wanted to highlight an example I found where, despite both conditions for a “peach-pair” having been met, the resulting sum is not prime:
22 + 232 = 533 = 13 × 41.
This led me to wonder whether or not there is a finite quantity of these “peach-pair” primes. Looking for an answer to that question may have to wait for another time.
Thanks for joining me to reflect on learning, teaching and doing mathematics. I hope it has been worthwhile.
Post script. I commonly make associations between events and the lyrics of songs. The playful title for the discovery I’ve recounted here was inspired by a song in an old film featuring Danny Kaye, who was a great favourite of my Dad:
“When it’s apple blossom time in Orange, New Jersey
We’ll make a peach of a pair”