Why the unit is not attributed to prime numbers, and when it was generally considered to be a number

My engineer friend recently surprised me. He said he was not sure whether the number 1 is prime or not. I was surprised because none of the mathematicians consider the unit simple.

The confusion begins with the definition given to a prime number: it is a positive integer that is divisible by only 1 and by itself . The number 1 is divided by 1, and it is divided by itself. But dividing yourself and 1 is not two different factors here. So is it a prime number or not? When I write the definition of a prime number, I try to eliminate this ambiguity: I am directly talking about the need for exactly two different conditions, dividing by 1 and by itself, or that a prime number must be an integer greater than 1. But why take such measures that exclude 1?

My mathematical education taught me that a good reason why 1 is not considered simple is the basic theorem of arithmetic. She claims that each number can be written as the product of primes in exactly one way. If 1 were simple, we would lose this uniqueness. We could write 2 as 1 × 2, or 1 × 1 × 2, or 1 ⁵⁹⁴⁸²⁷ × 2. Exception 1 from prime numbers eliminates this.

Initially, I planned to explain the basic theorem of arithmetic in an article and end it. But in fact, it is not so difficult to change the statement of the theorem to solve the problem with unity. In the end, my friend’s question sparked my curiosity: how did mathematicians settle on this definition of a prime? A quick search on Wikipedia showed that the unit was previously considered a prime number, but now it is not. But an article by Chris Caldwell and Yong Sungdemonstrates a slightly more complicated story. This can be understood from the very beginning of their article: “First, whether a number (especially a unit) is simple is a matter of determination, that is, a matter of choice, context and tradition, and not a matter of proof. However, definitions do not occur randomly; the choice is related to our use of mathematics and, especially in this case, our notation. ”

Caldwell and Xiong start with classical Greek mathematicians. They did not count 1 as a number, like 2, 3, 4, and so on. 1 was considered a number, and the number consisted of several digits. For this reason, 1 could not be simple - it is not even a number. 9th-century Arab mathematician al-Kindiwrote that this is not a number and, therefore, is not even or odd. For many centuries, the notion that a unit is the building block for compiling all numbers, but not the number itself, has prevailed.

In 1585, the Flemish mathematician Simon Stevin pointed out that in the decimal system there is no difference between 1 and any other numbers. In all respects, 1 behaves like any other quantity. Although not immediately, but this observation ultimately led mathematicians to accept 1 as any other number.

Until the end of the 19th century, some prominent mathematicians considered one simple and some not. As far as I can tell, this was not a cause of disagreement; for the most popular mathematical questions, the difference was not critical. Caldwell and Xiong cite G. H. Hardy as the last major mathematician who considers 1 to be simple (he explicitly indicated it as a prime in the first six editions of The Course of Pure Mathematics, published between 1908 and 1933, and in 1938 changed the definition and called 2 the least simple).

The article mentions, but does not understand in detail, the changes in mathematics, because of which 1 was excluded from the list of primes. In particular, one of the important changes was the development of sets outside the set of integers that behave as integers.

In the simplest example, we can ask if the number -2 is prime. The question may seem pointless, but it prompts us to express in words the unique role of unity among integers. The most unusual aspect of 1 is that its inverse is also an integer (the inverse of x is the number that, when multiplied by x, gives 1. For 2, the inverse of 1/2 is included in the set of rational or real numbers, but is not integer: 1/2 × 2 = 1). The number 1 turned out to be its own inverse number. No other positive integer has an inverse value in the set of integers. A number with an inverse value is called an invertible element.. The number −1 is also a reversible element in the set of integers: again, it is an invertible element for itself. We do not consider reversible elements as simple or compound, because you can multiply them by some other reversible elements without much change. Then we can assume that the number -2 is not so different from 2; in terms of multiplication. If 2 is prime, then −2 must be the same.

I carefully avoided in the previous paragraph the definition of a simple one due to the unfortunate fact that such a definition is not suitable for these large sets! That is, it is a little illogical, and I would choose another. For positive integers, every prime p has two properties:

It cannot be written as the product of two integers, none of which is a reversible element.

If the product m × n is divisible by p , then m or n must be divisible by p (for example, m = 10, n = 6, and p = 3.)

The first of these properties is how we could characterize primes but, unfortunately, an irreducible element is obtained here . The second property is a simple element . In the case of natural numbers, of course, the same numbers satisfy both properties. But this does not apply to every interesting set of numbers.

As an example, consider a set of numbers of the form a + b√ − 5 or a + ib√5 , where a and b  are integers and i  is the square root of −1. If you multiply the numbers 1 + √ − 5 and 1-√ − 5, you get 6. Of course, you also get 6, if you multiply 2 and 3, which are also in this set of numbers with b = 0. Each of the numbers 2, 3, 1 + √ − 5, and 1 − √ − 5 cannot be represented as the product of numbers that are not reversible elements (if you do not take my word for it, it is not too difficult to verify). But the product (1 + √ − 5) (1 − √ − 5) is divisible by 2, and 2 is not divisible by either 1 + √ − 5 or 1 − √ − 5 (again, you can check if you do not believe me ) Thus, 2 is an irreducible element, but not simple. In this set of numbers, 6 can be decomposed into irreducible elements in two different ways.

The above number, which mathematicians can call Z [√-5], contains two reversible elements: 1 and −1. But there are similar sets of numbers with an infinite number of reversible elements. Since such sets became objects of study, it makes sense to clearly distinguish between the definitions of reversible, irreducible, and simple elements. In particular, if there are sets of numbers with an infinite number of reversible elements, it becomes increasingly difficult to understand what we mean by unique factorization of numbers, unless you specify that invertible elements cannot be simple. Although I am not a historian of mathematics and do not deal with number theory and would like to read more about how this process happened, I think this is one of the reasons that Caldwell and Xiong consider the reason for the exclusion of 1 from prime numbers.

As often happens, my initial neat and concise answer to the question of why everything is arranged as it is, ultimately became only part of the problem. Thanks to my friend for asking a question and helping me learn more about the complex history of simplicity.