Can there be four square numbers in arithmetic progression? Can four numbers that are perfect powers be in arithmetic progression?
The simple answers are no and yes!
First I'll sketch the history of the first question, which developed from a problem studies by Islamic mathematicians more than a thousand years ago.
Then I'll outline how we know the answer to the second question and give an a examplle.
A little background: An arithmetic progression (AP) is when a series of numbers go up (or down) in equal steps, and this size of this step is called the common difference. For example 1, 4, 7, 10 are four numbers in AP, with common difference 3.
The congruum problem asks for triplets of square numbers that are in arithmetic progression, for example 1, 25, 49, whose common difference 24 is called the congruum.
Finding congruums (or congrua)? attracted the interest of medieval and later mathematicians, with several mathematical superstars getting in on the act.
According to Wikipedia the problem was first investigated in the 10th century, by Abu Jafar al-Khazin, who gave example such as the one above and found a general solution.
Things really hotted up in 1225 though, because the Holy Roman Emperor at that time organised a mathematical tournament. Can you imagine such a thing nowadays?
The prize was carried off by one Leonardo of Pisa, better known as Fibonacci, who wrote about the eponymous series 1, 1, 2, 3, 5, 8, ..., and about the Indo-Arabic numerals which replaced Roman numerals and to become the system used today.
Next on the stage is Fermat, lawyer and amateur mathematician who made a claim that got called Fermat's Last Theorem even though it wasn't really a theorem because at the time he never actually proved it (as far as we know). He also made a conjecture about squares in AP — that it is impossible to find a congruum that works for four squares, not just three. Could there be numbers that were in AP?
This challenge was taken up by Euler, another giant famous for many things, including introducing the symbol for the base of natural logarithm. He published a proof but by that time he had low vision and had a helper to tale his dictation, so some writers are not sure the published version is accurate.
Since that time there have been many proofs published but in the literature there are also authors who have commented flaws in earlier papers, and there is active research right up to the present day.
Now for the second problem. Can four numbers that are perfect powers be in arithmetic progression?
Absolutely. Start with if you like. The common difference is , so the next term would be . But, you will say, is not a perfect power. No, but is. So scale everything up by .
Now we get the four terms
We can use Python as a sort of super-calculator
1*73**2, 25*73**2, 49*73**2, 73**3
(5329, 133225, 261121, 389017)
133225 - 5329, 261121 - 133225,389017 - 261121
(127896, 127896, 127896)
and sure enough we have our AP.
Surprisingly we could in principle carry this as far as we like and get a million or a gazillion perfect powers in AP, although as you see the numbers grow dizzyingly fast; we have gone from 24 to 127896 in one step. So I imagine we would soon run out of resources and get to numbers too big to actually write dowm!