Written by Colin+ in dialogue, quadratics.

You want to know one of my favourite things about maths? It's the *connections*. Connections between things that don't, on the face of it, seem remotely connected. One just cropped up, and made me grin from ear to ear, so I thought I'd share it with you.

One of @standupmaths's favourite famous puzzles is the Magic Square Of Squares: can you place nine distinct square numbers in a three-by-three grid such that the diagonals, rows and columns all sum to the same number?

This is (to the best of my knowledge) an open problem: nobody has so far found an example, or a proof that no such magic square exists. (If one does, the numbers in it must be enormous).

I had a dabble with it, a few months ago; even though my level of number theory is almost certainly not enough to catch something nobody else has spotted, it's always good to give the muscles a bit of an exercise.

One of the rabbit-holes the puzzle led me down was the search for square numbers in arithmetic sequence — for example, 1, 25 and 49 have a common difference of 24. (I believe there is a theorem that the longest run of square in arithmetic series is three, but that there are infinitely many such triples). As I remember, the search for a MSoS is equivalent to finding three triples of squares in arithmetic series, all with the same common difference – just in case you wanted to carry on the exploration from where I left off.

Some time later, this horror-show of a tweet landed in my timeline:

Prime numbers? More spin than you can shake a stick at, a whirlwind of pupil engagement #ukedchat #primaryrocks #ks2 #matchchat #zedteaching pic.twitter.com/RGmHEWPLqI

— ZedTeach (@ZedTeach) October 7, 2017

(In fairness to @zedteach, there is a proviso that Demi is just looking up to 100 – but that "always" really bugs me.)

Of course, my loyal band of followers were quick to point out counterexamples — 120 (for which $119=7\times 17$ and $121 = 11^2$) was the first to spring to my mind, and 144 (one more than $11\times 13$, and one fewer than $29\times 5$) was a common answer.

Several good puzzles came out of this: @chrismaslanka wondered if there exists $k$ such that both $6k-1$ and $6k+1$ have at least three prime factors, which I'll leave you to explore.

In exploring it, @sxpmaths (Stuart Price) made the observation that $6n^2 + 5n – 1$ and $6n^2 + 5n + 1$ both factorise (as $(6n-1)(n+1)$ and $(3n+1)(2n+1)$, respectively), so making $n = 6m$ generates infinitely many multiples of six with neighbouring composites (although 120 and 144 are not caught by this.)

So I started investigating quadratics of the form $ax^2 + bx \pm 1$ to see whether they factorise. Quadratics of the form $Ax^2 + Bx + C$ factorise if $B^2 – 4AC$ is square, so we're looking for $a$ and $b$ such that $b^2 + 4a$ and $b^2 – 4a$ are both squares… oh look! Three squares in arithmetic sequence.

I haven't found the quadratic that corresponds to 120 yet – and part of me says "don't bother. You've found a Deeper Connection! Leave the trivial arithmetic for others."

Of course, other parts of my brain won't rest until I do find it… or one of you, loyal readers, tells me what it is.

* Thanks to @grey_matter for spotting that 119 is not, in fact, $7 \times 13$.