Chapter 4. "Of the Nature of whole Numbers, or Integers, with respect to their Factors."
Part I. Section I. Chapter 4. “Of the Nature of whole Numbers, or Integers, with respect to their Factors.”
37 We have observed that a product is generated by the multiplication of two or more numbers together, and that these numbers are called factors. Thus, the numbers \(a\), \(b\), \(c\), \(d\), are the factors of the product \(abcd\).
38 If, therefore, we consider all whole numbers as products of two or more numbers multiplied together, we shall soon find that some of them cannot result from such a multiplication, and consequently have not any factors; while others may be the products of two or more numbers multiplied together, and may consequently have two or more factors. Thus 4 is produced by 2 · 2; 6 by 2 · 3; 8 by 2 · 2 · 2; 27 by 3 · 3 · 3; and 10 by 2 · 5, etc.
39 But on the other hand, the numbers 2, 3, 5, 7, 11, 13, 17, etc. cannot be represented in the same manner by factors, unless for that purpose we make use of unity, and represent 2, for instance, by 1 · 2. But the numbers which are multiplied by 1 remaining the same, it is not proper to reckon unity as a factor.
All numbers, therefore, such as 2, 3, 5, 7, 11, 13, 17, etc. which cannot be represented by factors, are called simple, or prime numbers; whereas others, as 4, 6, 8, 9, 10,
12, 14, 15, 16, 18, etc. which may be represented by factors, are called composite numbers.1
40 Simple or prime numbers deserve therefore particular attention, since they do not result from the multiplication of two or more numbers. It is also particularly worthy of observation, that if we write these numbers in succession as they follow each other, thus,
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, etc.
we can trace no regular order; their increments being sometimes greater, sometimes less; and hitherto no one has been able to discover whether they follow any certain law or not.
41 All composite numbers, which may be represented by factors, result from the prime numbers above mentioned; that is to say, all their factors are prime numbers. For, if we find a factor which is not a prime number, it may always be decomposed and represented by two or more prime numbers. When we have represented, for instance, the number 30 by 5 · 6, it is evident that 6 not being a prime number, but being produced by 2 · 3, we might have represented 30 by 5 · 2 · 3, or by 2 · 3 · 5; that is to say, by factors which are all prime numbers.
42 If we now consider those composite numbers which may be resolved into prime factors, we shall observe a great difference among them; thus we shall find that some have only two factors, that others have three, and others a still greater number. We have already seen, for example, that
4 is the same as 2 · 2,
6 is the same as 2 · 3,
8 is the same as 2 · 2 · 2,
9 is the same as 3 · 3,
10 is the same as 2 · 5,
12 is the same as 2 · 2 · 3,
14 is the same as 2 · 7,
15 is the same as 3 · 5,
43 Hence, it is easy to find a method for analysing any number, or resolving it into its simple factors. Let there be proposed, for instance, the number 360; we shall represent it first by 2 · 180. Now 180 is equal to 2 · 90, and
90 is the same as 2 · 45,
45 is the same as 3 · 15,
15 is the same as 3 · 5.
So that the number 360 may be represented by these simple factors,
2 · 2 · 2 · 3 · 3 · 5;
since all these numbers multiplied together produce 360.
44 This shows, that prime numbers cannot be divided by other numbers; and, on the other hand, that the simple factors of compound numbers are found most conveniently, and with the greatest certainty, by seeking the simple, or prime numbers, by which those compound numbers are divisible. But for this division is necessary; we shall therefore explain the rules of that operation in the following chapter.
- Leonhard Euler. Elements of Algebra. Translated by Rev. John Hewlett. Third Edition. Longmans, Hurst, Rees, Orme, and Co. London. 1822.
- Leonhard Euler. Vollständige Anleitung zur Algebra. Mit den Zusätzen von Joseph Louis Lagrange. Herausgegeben von Heinrich Weber. B. G. Teubner. Leipzig and Berlin. 1911. Leonhardi Euleri Opera omnia. Series prima. Opera mathematica. Volumen primum.
According to Euclid’s definitions, 1 (unity) is not considered a number, and therefore is not considered a prime number. “Greek mathematicians tend to conceive of number (arithmos) as a plurality of units. Perhaps a better translation, without our deeply entrenched notions, would be ‘count’.” Mendell, Henry, “Aristotle and Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2019/entries/aristotle-mathematics/. Section 10. Unit (monas) and Number (arithmos)
According to Euler’s terminology, 1 is described as unity and is not considered either prime or composite: thus a positive integer is either unity, prime, or composite. Modern mathematics follows the same convention as Euler: 1 is a positive integer with exactly one positive factor; a prime number is a positive integer with exactly two positive factors; and a composite number is a positive integer with more than two positive factors. Hewlett’s translation - unlike the German original - adds material treating 1 as a prime number, which I have removed. ↩