Mathematics is often about uncovering patterns. For example, certain areas of topology revolve around categorizing knots or geometric shapes, and number theory explores properties such as the distribution of prime numbers. If we restrict ourselves to somewhat simpler relationships, we can observe a pattern with the numbers 5 and 6 that was recognized by the Babylonians millennia ago: the square of 5 is 25, which ends with 5; the square of 25 is 625, which ends with 25; and the square of 625 is 390,625, which ends with 625. What seems like a fun gimmick made popular by mathematician Maurice Kraitchik in 1942 leads to one of the most important number systems in mathematics—and one of the strangest.
If you play with the number 6, the result is not quite as impressive, but here, too, a pattern emerges: 6 squared gives 36; 36 squared gives 1,296. Although 36 no longer appears in the sequence of digits, the result always ends with 6. In general, numbers whose square ends with the same digit or digits as the number itself are called automorphic. There are an infinite number of these: 0, 1, 5, 6, 25, 76, 376, and so on. As it turns out, aside from 0 and 1, all automorphic numbers end with either 5 or 6.
The number 5 is particularly exciting, however. Not only is it automorphic, but also its square and the square of its square are automorphic. This naturally raises the question of whether this sequence of automorphic numbers continues indefinitely. In other words, does the repeated squaring of 5 always produce an automorphic number?
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As it turns out, that is not the case:
Credit: Spektrum der Wissenschaft, styled by Amanda Montañez
So the pattern seems to collapse after the third square: 390,6252 results in 152,587,890,625. Therefore 390,625 cannot be automorphic because the number is not completely contained in its square.
But if you look carefully, you can see that at least the last five digits appear in the square number, namely 90,625. And if you square this number, you get: 8,212,890,625. Therefore 90,625 is an automorphic number!
That means you can go on and calculate the square of 8,212,890,625. The result is huge, but it turns out that 8,212,890,625 is also automorphic because its square is 67,451,572,418,212,890,625.
You can continue this procedure: successively square all the numbers, and if they are not automorphic, continue the calculations with the last digits that are repeated. This results in the following sequence of numbers:
5
25
625
90,625
8,212,890,625
18,212,890,625
918,212,890,625
As you can see, this results in an ever larger automorphic number. In fact, this procedure can be continued to infinity—in the end, the result is an infinitely large number that is completely automorphic (that is, an infinitely large number whose square corresponds to itself: n2 = n). Even if you cannot write down that infinitely large number, its last digits are known: ...918,212,890,625.
The fact that there is such a “fixed point” in infinity is astonishing in itself. The fact that at least the last digits of this number can be specified precisely is even more astonishing.
It is not immediately obvious that this procedure can be continued infinitely often. After all, at some point you could come across a number that is no longer automorphic. And anyway—what is an infinite number like ...67,451,572,418,212,890,625 supposed to represent? How is it different from a value such as ...11111111111? After all, both numbers are infinite.
A New Number System Is Born
In the late 19th century mathematician Kurt Hensel developed the concept of so-called p-adic numbers. These are numbers that have an infinite number of digits before the decimal point—in contrast to ordinary real numbers, which continue indefinitely after the decimal point, such as π = 3.14159.... Even if this sounds extremely unusual at first, you can calculate with p-adic numbers in the same way as ordinary real numbers.
To see this, consider a somewhat unusual representation of the real numbers. Every real number can also be expressed as an infinite sum. For example, π = 3 x 100 + 1 x 10-1 + 4 x 10-2 + 1 x 10-3 + 5 x 10-4 + 9 x 10-5 + ...
The p-adic numbers can also be represented as an infinite series but with positive exponents. So ...890625 = 5 x 100 + 2 x 101 + 6 x 102 + 0 x 103 + 9 x 104 + 8 x 105 + .... In this way, it becomes clearer how you would calculate with these strange numbers. For example, ...111111 + ...22222 = ...33333. The p-adic numbers can also be divided and multiplied.
The last two operations can lead to problems with automorphic numbers such as ...890,625, however. As already mentioned, this number corresponds to its square, so the following applies: n2 = n.
If you convert this quadratic equation, the result is: n2 – n = n x (n – 1) = 0. If a product of two factors (here n and n – 1) results in 0, then at least one of the factors must be 0. This is only the case, however, if n = 0 or n = 1. With p-adic numbers, n can also have a value other than 0 or 1, such as ...890,625, for example, and still fulfill the above equation. This means that with p-adic numbers, the product of two numbers that are both not equal to 0 can still result in 0.
Division by Zero
Such “zero divisors” pose a problem even in simple calculations. Suddenly you have to be extremely careful when dividing to avoid accidentally dividing a number by 0. This can be seen in the following example: Assume that a and b are p-adic numbers that are not equal to 0 and that a x b = 0. If you want to solve the equation 2⁄a = b x (1 + x) for x, you would usually divide both sides of the equation by b first. Because the product of a and b is 0, however, you would divide the left-hand term by 0. The equation therefore cannot be solved in this way.
As it turns out, such problematic zero divisors can be avoided. In case you were wondering about the name of the number system, the p stands for prime number. The p-adic numbers that I have presented are actually “10-adic” numbers, however, which are defined in base 10. Because 10 is not a prime number, such unpleasant zero divisors occur. But if you were to look at the 3-adic numbers, for example, which are represented by a sum of the form x0 x 30 + x1 x 31 + x2 x 32 + x3 x 33 + x4 x 34 + x5 x 35 + ... (where the coefficients xi = 0, 1 or 2), you would not find any zero divisors. And thus, p-adic numbers where p is really a prime number do not contain any completely automorphic values that fulfill n2 = n, apart from ...00000 and ...00001 (0 and 1).
Although the p-adic numbers seem extremely complicated at first glance, they are widely used. In fact, number theorists use these strange values for most of their work. p-adic numbers are “far removed from our everyday intuitions,” mathematician Peter Scholze told Quanta magazine. “Now I find real numbers much, much more confusing than p-adic numbers. I’ve gotten so used to them that now real numbers feel very strange.”
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.