Prime Numbers

Summary: Integers greater than 1 that have no factors other than 1 and themselves; the building blocks from which all composite numbers are constructed by multiplication.

Sources: chapter-1.1.4, chapter-1.1.6

Last updated: 2026-04-24

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Definition

A prime number (Euler: simple number) is one that cannot be expressed as a product of two or more smaller positive integers. Equivalently, its only divisors are 1 and itself (source: chapter-1.1.4).

Unity (1) is not counted as prime.

The Sequence of Primes

$2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,\dots$

Euler observes: “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” (source: chapter-1.1.4).

Role in Factorisation

Every composite number decomposes into a product of primes (the Fundamental Theorem of Arithmetic, implicit in Euler’s treatment). Any non-prime factor can itself be broken into primes:

$360=2^3\times 3^2\times 5$

Once a number is expressed in prime factors, all its divisors can be listed by forming products of subsets of those factors — see ch1.1.6-properties-integers-divisors (source: chapter-1.1.4, chapter-1.1.6).

Composite Numbers

Numbers such as $4,6,8,9,10,12,\dots$ that have factors beyond 1 and themselves. Key examples:

Composite	Prime factorisation
4	$2^2$
6	$2\times 3$
12	$2^2\times 3$
30	$2\times 3\times 5$
360	$2^3\times 3^2\times 5$

Related pages

• ch1.1.4-integers-and-factors
• ch1.1.6-properties-integers-divisors
• ch1.1.5-division-simple-quantities