The Search for the Largest Square Number

Just three weeks ago, the Great Internet Search for Squares (GISS) turned up the massive square number $(5^{7457329477993})^2$ , dwarfing the previously largest known square.

When a new largest prime number is discovered, it receives much attention in the mathematical world, and it even crops up in news for the general public. But with the primes stealing the spotlight, the search for the largest square number garners considerably less interest. Nevertheless, it is a rigorous undertaking with a rich history and a promising future.

It is clear from Pythagoras’ writings that certain small squares were known in antiquity. In Foundations of Mathematics, he writes (translated from ancient Greek) $3^2 + 4^2 = 5^2$ , and in a later text, $5^2 + 12^2 = 13^2$ . Some historians claim that he indirectly references larger squares, even as large as $100^2$ , but there is no consensus on whether the Greeks were familiar with squares larger than $13^2$ .

The first major breakthrough in the study of large squares is owed to Daniel Bernoulli, who spent much of his career tabulating square numbers. His table would become a familiar resource for number theorists who needed to know, in the heat of a difficult problem, whether $19334^2$ was in fact a square number.

Bernoulli’s table extended to $26000^2$ , though this is not without controversy. Written in Bernoulli’s famously illegible hand, the exponent appears to some to be a 7 rather than a 2. While $26000^2$ is indeed a square number, $26000^7$ is not. A small subset of the math community argues that he should only be credited with discovering up to $25999^2$ (the first square smaller than $26000^2$ ).

Regardless of the status of Bernoulli’s Last Square—as conspiracy theorists have termed it—he increased the quantity of known squares by multiple orders of magnitude. But Bernoulli’s seminal achievement came later in life, when he supplied a proof that there are infinite square numbers. The original proof is beyond the scope of this article, but in essence, his reasoning was as follows.

Say there were only finite squares. Then we could call the number of squares $n$ . We could write the first as $a_1^2$ , the second as $a_2^2$ , and so on until the last as $a_n^2$ , where $a_1, a_2, \dots, a_n$ are distinct integers. But when we multiply all those squares, we get $a_1^2 a_2^2 \dots a_n^2$ , which is really just $(a_1 a_2 \dots a_n)^2$ , another square. This new square is larger than each of $a_1^2, a_2^2, \dots, a_n^2$ , so it is not among them. But this means we have found an $(n+1)^\text{th}$ square, whereas we had assumed there were only $n$ . Since we have been led to a contradiction, our assumption that there were only finitely many squares must have been wrong.

Perhaps as Bernoulli finished his proof, he glimpsed for a moment the competition that would arise around the search for squares. With no limit on the size of squares, there would always be a greater one to discover—and no shortage of fame for its discoverer. Though the GISS was not yet active—computers were still only a dream of Charles Babbage—the hunt for ever-larger squares was on.

In 1788, the English mathematician Leonard Euler provided $46656^2$ , shattering the previous record. Two years later, he gave $(50000 + \phi(1335))^2$ , where $\phi$ is Euler’s totient function, curiously a device related to the prime numbers.

In 1817, Pierre Fermat proposed $(221999)(222000)$ , but this was later shown not to be square by the clever argument, per Bernhard Riemann, that it sits between two consecutive squares: $221999^2$ and $222000^2$ . In formulating this refutation, Riemann realized that he had in fact discovered two new largest squares.

A few years later, Riemann’s student, the child prodigy Friedrich Gauss, asked his mentor what the largest known square was. Riemann replied with his own discovery: $222000^2$ .

“Oh yeah?” Gauss retorted. “Two hundred and twenty-two thousand plus one, squared. Beat that.”

Riemann was shocked, but he managed to stammer, “Two hundred and twenty-two thousand plus two, squared.”

The young Gauss thought for a moment, then responded, “Two hundred and twenty-two thousand plus a million billion million, squared.”

Riemann had no choice but to concede defeat. Gauss’ square, which written out has 43 digits, was the largest known square for close to a decade.

The larger the numbers get, the more difficult it is to find squares. Like the primes, the squares thin out. Whereas 25, 36, and 49 are all within counting distance of each other, there are almost two million consecutive non-squares following the millionth square. With such mind-boggling gaps between squares, it is not surprising that square enthusiasts turned to computers for help.

Today, all of the largest squares are owed to the recently newsworthy GISS. The GISS software runs on millions of personal computers around the world, trawling through enormous numbers and doing complicated calculations to look for squares. When a new largest square is found, the owner of the computer that turned it up gets credit for its discovery. Anyone with a laptop can participate, and installing the GISS software takes only a few minutes.

So what are you waiting for? Don’t be a square. Start searching today!

Written: 11/14/2017

The Search for the Largest Square Number

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