An engineer friend of mine recently surprised me by saying he wasn’t sure whether the number 1 was prime or not. I was surprised because among mathematicians, 1 is universally regarded as non-prime.
The confusion begins with this definition a person might give of “prime”: a prime number is a positive whole number that is only divisible by 1 and itself. The number 1 is divisible by 1, and it’s divisible by itself. But itself and 1 are not two distinct factors. Is 1 prime or not? When I write the definition of prime in an article, I try to remove that ambiguity by saying a prime number has exactly two distinct factors, 1 and itself, or that a prime is a whole number greater than 1 that is only divisible by 1 and itself. But why go to those lengths to exclude 1?
My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1594827×2. Excluding 1 from the primes smooths that out.
My original plan of how this article would go was that I would explain the fundamental theorem of arithmetic and be done with it. But it’s really not so hard to modify the statement of the fundamental theorem of arithmetic to address the 1 problem, and after all, my friend’s question piqued my curiosity: how did mathematicians coalesce on this definition of prime? A cursory glance around some Wikipedia pages related to number theory turns up the assertion that 1 used to be considered prime but isn’t anymore. But a paper by Chris Caldwell and Yeng Xiong shows the history of the concept is a bit more complicated. I appreciated this sentiment from the beginning of their article: “First, whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.”
Caldwell and Xiong start with classical Greek mathematicians. They did not consider 1 to be a number in the same way that 2, 3, 4, and so on are numbers. 1 was considered a unit, and a number was composed of multiple units. For that reason, 1 couldn’t have been prime — it wasn’t even a number. Ninth-century Arab mathematician al-Kindīwrote that it was not a number and therefore not even or odd. The view that 1 was the building block forall numbers but not a number itself lasted for centuries.
In 1585, Flemish mathematician Simon Stevin pointed out that when doing arithmetic in base 10, there is no difference between the digit 1 and any other digits. For all intents and purposes, 1 behaves the way any other magnitude does. Though it was not immediate, this observation eventually led mathematicians to treat 1 as a number, just like any other number.
Through the end of the 19th century, some impressive mathematicians considered 1 prime, and some did not. As far as I can tell, it was not a matter that caused strife; for the most popular mathematical questions, the distinction was not terribly important. Caldwell and Xiong cite G. H. Hardy as the last major mathematician to consider 1 to be prime. (He explicitly included it as a prime in the first six editions of A Course in Pure Mathematics, which were published between 1908 and 1933. He updated the definition in 1938 to make 2 the smallest prime.)
The article mentions but does not delve into some of the changes in mathematics that helped solidify the definition of prime and excluding 1. Specifically, one important change was the development of sets of numbers beyond the integers that behave somewhat like integers.
In the very most basic example, we can ask whether the number -2 is prime. The question may seem nonsensical, but it can motivate us to put into words the unique role of 1 in the whole numbers. The most unusual aspect of 1 in the whole numbers is that it has a multiplicative inverse that is also an integer. (A multiplicative inverse of the number x is a number that when multiplied by x gives 1. The number 2 has a multiplicative inverse in the set of the rational or real numbers, 1/2: 1/2×2=1, but 1/2 is not an integer.) The number 1 happens to be its own multiplicative inverse. No other positive integer has a multiplicative inverse within the set of integers.* The property of having a multiplicative inverse is called being a unit. The number -1 is also a unit within the set of integers: again, it is its own multiplicative inverse. We don’t consider units to be either prime or composite because you can multiply them by certain other units without changing much. We can then think of the number -2 as not so different from 2; from the point of view of multiplication, -2 is just 2 times a unit. If 2 is prime, -2 should be as well.
I assiduously avoided defining prime in the previous paragraph because of an unfortunate fact about the definition of prime when it comes to these larger sets of numbers: it is wrong! Well, it’s not wrong, but it is a bit counterintuitive, and if I were the queen of number theory, I would not have chosen for the term to have the definition it does. In the positive whole numbers, each prime number p has two properties:
The number p cannot be written as the product of two whole numbers, neither of which is a unit.
Whenever a product m×n is divisible by p, then m or n must be divisible by p. (To check out what this property means on an example, imagine that m=10, n=6, and p=3.)
The first of these properties is what we might think of as a way to characterize prime numbers, but unfortunately the term for that property is irreducible. The second property is called prime. In the case of positive integers, of course, the same numbers satisfy both properties. But that isn’t true for every interesting set of numbers.
As an example, let’s look at the set of numbers of the form a+b√-5, or a+ib√5, where a and b are both 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 in this set of numbers as well, with b=0. Each of the numbers 2, 3, 1+√-5, and 1-√-5 cannot be broken down further and written as the product of numbers that are not units. (If you don’t take my word for it, it’s not too difficult to convince yourself.) But the product (1+√-5)(1-√-5) is divisible by 2, and 2 does not divide either 1+√-5 or 1-√-5. (Once again, you can prove it to yourself if you don’t believe me.) So 2 is irreducible, but it is not prime. In this set of numbers, 6 can be factored into irreducible numbers in two different ways.
The number set above, which mathematicians might call Z[√-5] (pronounced "zee adjointhe square root of negative five" or "zed adjointhe square root of negative five, pip pip, cheerio" depending on what you like to call the last letter of the alphabet), has two units, 1 and -1. But there are similar number sets that have an infinite number of units. As sets like this became objects of study, it makes sense that the definitions of unit, irreducible, and prime would need to be carefully delineated. In particular, if there are number sets with an infinite number of units, it gets more difficult to figure out what we mean by unique factorization of numbers unless we clarify that units cannot be prime. While I am not a math historian or a number theorist and would love to read more about exactly how this process took place before speculating further, I think this is one development Caldwell and Xiong allude to that motivated the exclusion of 1 from the primes.
As happens so often, my initial neat and tidy answer for why things are the way they are ended up being only part of the story. Thanks to my friend for asking the question and helping me learn more about the messy history of primality.
*This sentence was edited after publication to clarify that no other positive integer has a multiplicative inverse that is also an integer.
The views expressed are those of the author(s) and are not necessarily those of Scientific American.
ABOUT THE AUTHOR(S)
Evelyn Lamb is a freelance math and science writer based in Salt Lake City, Utah.Follow Evelyn Lamb on Twitter
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FAQs
When did 1 stop being a prime number? ›
By the early 20th century, mathematicians began to agree that 1 should not be listed as prime, but rather in its own special category as a "unit".
Why is 1 and 0 not a prime number? ›By definition, Prime numbers are those natural numbers that are divisible solely by the unity (1) and themselves. So, 0 is divided by every natural number. Number 1 has positive divisors as 1 and itself and it must have only two positive factors.
Who decided that 1 is not a prime number? ›Caldwell and Xiong start with classical Greek mathematicians. They did not consider 1 to be a number in the same way that 2, 3, 4, and so on are numbers. 1 was considered a unit, and a number was composed of multiple units. For that reason, 1 couldn't have been prime — it wasn't even a number.
Why is 2 A prime number and 1 isn t? ›Proof: The definition of a prime number is a positive integer that has exactly two distinct divisors. Since the divisors of 2 are 1 and 2, there are exactly two distinct divisors, so 2 is prime. Rebuttal: Because even numbers are composite, 2 is not a prime. Reply: That is true only for all even numbers greater than 2.
What was America's first prime number? ›Prime numbers are numbers that have only 2 factors: 1 and themselves. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11.
What is the largest prime number? ›The largest known prime number (as of February 2023) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.
What is the smallest prime number? ›A prime number must have exactly two factors (1 and the number itself). Therefore, the number 2 is the lowest prime number as its only factors are 1 and 2. Q.
Why 2 is not a prime number? ›Yes, 2 is a prime number.
According to the definition of prime numbers, any whole number which has only 2 factors is known as a prime number. Now, the factors of 2 are 1 and 2. Since there are exactly two factors of 2, it is a prime number.
2 is the only even prime number.
Why is 0 not prime? ›For an integer to be prime it must be greater than 1, and the only integers that divide into it exactly are 1 and itself such as 3 and 13, etc. 0 is less than 1 so can't be prime. Composite integers are those that are the products of primes such as 6 = 2x3.
What is the first non prime number? ›
Definition: A prime number is a whole number with exactly two integral divisors, 1 and itself. The number 1 is not a prime, since it has only one divisor. The number 4 is not prime, since it has three divisors ( 1 , 2 , and 4 ), and 6 is not prime, since it has four divisors ( 1 , 2 , 3 , and 6 ).
Are all odd numbers prime? ›No, every odd number is not a prime number. Example: 9 is an odd number having factors 1, 3 and 9 and is not a prime number.
How do you know if a number is prime? ›How do you know a prime number? If a number has only two factors 1 and itself, then the number is prime.
What are prime numbers rules? ›A prime number is a number which has just two factors: itself and 1. Or in other words it can be divided evenly only by itself and 1. For instance, 3 is a prime number because it can be divided evenly only by itself and one. On the other hand, 6 can be divided evenly by 1, 2, 3 and 6.
Why is 9 not a prime number? ›9 is not a prime number. It can be divided by 3 as well as 1 and 9. The prime numbers below 20 are: 2, 3, 5, 7, 11, 13, 17, 19. Don't forget: the number 1 is not a prime number as it only has one factor.
Why is 69 not a prime number? ›The number 69 is a composite number. Its factors are 1, 3, 23, and 69. Because 69 has more than two factors, it is a composite number rather than a prime number.
Why is 27 not a prime number? ›Is 27 a prime number? No. 27 is divisible by other numbers (3 and 9), so it is not prime. The factors of 27 are 1, 3, 9, and 27, so it is not prime.
What can go into 24? ›There are a total of eight factors of 24, they are 1, 2, 3, 4, 6, 8, 12 and 24.
Are there infinite prime numbers? ›The number of primes is infinite. The first ones are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 and so on. The first proof of this important theorem was provided by the ancient Greek mathematician Euclid. His proof is known as Euclid's theorem.
Why is infinity not a number? ›Infinity is not a number, but if it were, it would be the largest number. Of course, such a largest number does not exist in a strict sense: if some number n n n were the largest number, then n + 1 n+1 n+1 would be even larger, leading to a contradiction. Hence infinity is a concept rather than a number.
Can negative numbers be prime? ›
Answer One: No.
By the usual definition of prime for integers, negative integers can not be prime. By this definition, primes are integers greater than one with no positive divisors besides one and itself. Negative numbers are excluded.
Number 11 is a prime number because it doesn't have proper factors. In other words, the only factors of 11 are 1 and itself.
What is the largest 2 digit prime number? ›97 is: the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101.
What is the smallest twin prime number? ›Answer: The twin primes between 1 and 100 are; (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).
Why isn't 8 a prime number? ›The number 8 is divisible by 1, 2, 4, 8. For a number to be classified as a prime number, it should have exactly two factors. Since 8 has more than two factors, i.e. 1, 2, 4, 8, it is not a prime number.
What is 100th prime number? ›100th prime is 541. 541 is 100th prime.
Is √ 2 a prime number? ›Prime numbers are Natural numbers. Since √2 is strictly a Real number, it can't be a prime like the other primes we have.
What is the smallest perfect number? ›perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128.
Why is 57 not a prime number? ›No, 57 is not a prime number since it has more than 2 factors and they are 1, 3, 19, and 57. Thus, by the definition of prime numbers, we can say that 57 cannot be a prime number.
Which number is considered the magic number? ›It is 1729. Discovered by mathemagician Srinivas Ramanujan, 1729 is said to be the magic number because it is the sole number which can be expressed as the sum of the cubes of two different sets of numbers.
Why is 6 a prime number? ›
No, 6 is not a prime number. The number 6 is divisible by 1, 2, 3, 6. For a number to be classified as a prime number, it should have exactly two factors. Since 6 has more than two factors, i.e. 1, 2, 3, 6, it is not a prime number.
Is Pi is a prime number? ›...
Pi-Prime.
| decimal digits | discoverer | date |
|---|---|---|
| 78073 | E. W. Weisstein | Jul. 13, 2006 |
| 613373 | A. Bondrescu | May 29, 2016 |
2 is a prime number because it has two positive divisors 1 and 2 which is satisfying the definition of a prime number. 2 is not a composite number because it has not divisors which are other than 1 or itself. Hence, 2 is a prime number but not a composite number.
Is 123456789 A prime numbers? ›This number is a composite.
Why is 45 a prime number? ›No, 45 is not a prime number. The number 45 is divisible by 1, 3, 5, 9, 15, 45. For a number to be classified as a prime number, it should have exactly two factors. Since 45 has more than two factors, i.e. 1, 3, 5, 9, 15, 45, it is not a prime number.
Why is 47 a prime number? ›The number 47 has only two factors, i.e. 1 and 47 so it is a prime number.
Why is 99 not a prime number? ›No, 99 is not a prime number. The number 99 is divisible by 1, 3, 9, 11, 33, 99. For a number to be classified as a prime number, it should have exactly two factors. Since 99 has more than two factors, i.e. 1, 3, 9, 11, 33, 99, it is not a prime number.
Why 51 is not a prime number? ›No, 51 is not a prime number since it has more than two factors. The factors of 51 can be written as 1, 3, 17, 51.
What is the smallest odd number? ›Smallest odd whole number is 1 .
How rare is a prime number? ›Prime numbers are abundant at the beginning of the number line, but they grow much sparser among large numbers. Of the first 10 numbers, for example, 40 percent are prime — 2, 3, 5 and 7 — but among 10-digit numbers, only about 4 percent are prime.
What is the fastest way to find a prime number? ›
Prime sieves are almost always faster. Prime sieving is the fastest known way to deterministically enumerate the primes.
What numbers Cannot be prime? ›Zero and 1 are not considered prime numbers. Except for 0 and 1, a number is either a prime number or a composite number. A composite number is defined as any number, greater than 1, that is not prime.
What is the logic of prime? ›A natural number is said to be prime if it is only divisible by itself and 1. In short, a prime number has only two factors that are 1 and the number itself. The numbers that are not prime are called composite numbers. A prime number can be written as a product of only two numbers.
How many prime numbers exist? ›...
| Prime Numbers between 1 and 100 | |
|---|---|
| Prime numbers between 1 and 10 | 2, 3, 5, 7 |
| Prime numbers between 30 and 40 | 31, 37 |
| Prime numbers between 40 and 50 | 41, 43, 47 |
15 is not an example of a prime number because it can be divided by 5 and 3 as well as by itself and 1. 15 is an example of a composite number because it has more than two factors.
Why is it impossible to find prime numbers? ›First, except for the number 2, all prime numbers are odd, since an even number is divisible by 2, which makes it composite.
Why is 7 not a prime number? ›Seven is a prime number because it doesn't have proper factors. In other words, the only factors of 7 are 1 and itself.
Why is 1 and 9 not a prime number? ›A natural number is called a prime number if it is greater than 1, and it doesn't have proper factors. For example, the prime numbers less than 9 are 2, 3, 5, and 7. A composite number is a natural number that has proper factors.
Is one number in the 90s prime? ›97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number and the only prime in the nineties.
Why was 2 a prime number? ›Yes, 2 is a prime number.
According to the definition of prime numbers, any whole number which has only 2 factors is known as a prime number. Now, the factors of 2 are 1 and 2. Since there are exactly two factors of 2, it is a prime number.
Is 2 a prime number yes or no? ›
The number 2 is prime. (It is the only even prime.)
What is the trick for prime numbers? ›To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can't be a prime number. If you don't get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).
Why is 45 not a prime number? ›Why is 45 not a prime number? 45 is not a prime number since it has more than two factors. As we know, a prime number has only two factors, 1 and the number itself.
Are there any even prime numbers? ›The unique even prime number 2. All other primes are odd primes. Humorously, that means 2 is the "oddest" prime of all.
Why is 34 not a prime number? ›No, 34 is not a prime number. The number 34 is divisible by 1, 2, 17, 34. For a number to be classified as a prime number, it should have exactly two factors. Since 34 has more than two factors, i.e. 1, 2, 17, 34, it is not a prime number.