General · EdgepediaSeptember 17, 20265 min read

3

3 (three) is the smallest odd prime number and the second prime overall, and it is the only prime that is one more than another prime, namely 2.¹ Because 2 and 3 are the only consecutive primes, 3 anchors several distinctive facts in elementary number theory: it is the smaller member of the first two twin-prime pairs, (3, 5) and (5, 7), and the only value of n for which the triple (n, n + 2, n + 4) consists entirely of primes.²

Key fact	Detail
Primality	Smallest odd prime; the only prime one more than another prime (2)¹
Consecutive primes	2 and 3 are the only consecutive primes; every other pair of primes differs by at least 2²
Divisibility test	An integer is divisible by 3 if and only if its digit sum is³
Sequence membership	Appears among the odd numbers, primes, Fibonacci numbers, Lucas numbers, and triangular numbers¹
Partitions	Exactly three partitions of 3: {1, 1, 1}, {1, 2}, and {3}¹
Approximate values	$\sqrt{3} \approx 1.73205080$ ; $\log_{2} 3 \approx 1.58496250$ ¹
Representations	11 in binary, 10 in ternary, 111 in negabinary¹

Arithmetic character

As a prime, 3 has exactly two positive divisors, 1 and itself, and it is the building block of the smallest odd numbers in the prime sequence 2, 3, 5, 7, 11, 13, ...¹^•⁴ Its primality interacts with parity in a way no other prime shares: since 2 and 3 are consecutive, every prime after 3 differs from every other prime by at least 2, and the twin-prime pairs (3, 5) and (5, 7) are the only overlapping twin pairs.² The same divisibility-by-3 obstruction explains why the prime triple (n, n + 2, n + 4) exists only for n = 3, giving (3, 5, 7): among any three numbers spaced by 2, one is divisible by 3.²

The number 3 also appears in the classical sequences: it is the third odd number, the second prime, the fourth Fibonacci number, the third Lucas number, and the second triangular number.¹ In Pascal's triangle it occurs twice, and in Lozanić's triangle four times.¹

The digit-sum divisibility test

The best-known practical fact about 3 is its divisibility test: an integer is divisible by 3 if and only if the sum of its digits is divisible by 3.³ The reason is that 10 is congruent to 1 modulo 3, so each power of 10 in a decimal expansion contributes only its digit:

$n = \sum_{i=0}^{k} d_{i} \cdot 10^{i} \equiv \sum_{i=0}^{k} d_{i} \pmod{3}.$

The same argument works for 9, since 10 ≡ 1 (mod 9) as well, and the test can be iterated when the digit sum itself is large; for 8,675,038,695 the digit sum is 57, whose digit sum is 12, so the original number is divisible by 3.³ The rule is a special case of modular arithmetic, the system of congruences published by Carl Friedrich Gauss in 1801, in which two integers are congruent modulo n when they leave the same remainder on division by n.⁵

Powers, roots, and logarithms

The powers of 3 begin 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, forming the sequence of powers of three.¹ Its square root is $\sqrt{3} \approx 1.73205080$ , and the base-2 logarithm $\log_{2} 3 \approx 1.58496250$ measures how many bits are needed to distinguish three states, a quantity that appears in information theory and computer science.¹

Behavior in other number systems

In ordinary base-10 notation 3 is a single digit, but its representation varies with the base: it is 11 in binary, 10 in ternary (the base it generates), and 111 in negabinary, since $(-2)^{2} + (-2)^{1} + (-2)^{0} = 4 - 2 + 1 = 3$ .¹

In algebraic number theory, 3 is prime in the ordinary integers $\mathbb{Z}$ but composite in some quadratic integer rings. In $\mathbb{Z}[\sqrt{3}]$ it factors as $(\sqrt{3})^{2}$ , and in the Eisenstein integers $\mathbb{Z}[\omega]$ , where $\omega = -\tfrac{1}{2} + \tfrac{\sqrt{-3}}{2}$ is a complex cube root of unity, it factors as $(-1)(1 + 2\omega)^{2}$ .¹ The ring $\mathbb{Z}[\omega]$ has six units, more than any other imaginary quadratic integer ring, a distinction that traces directly back to the role of 3 in defining $\omega$ .¹

Primality testing modulo 3 is not foolproof: composite numbers called Fermat pseudoprimes to base 3, beginning 91, 121, 286, 671, and 703, satisfy the base-3 Fermat condition without being prime.¹

Sums of three triangular numbers

The triangular numbers 1, 3, 6, 10, 15, ... are the sums 1 + 2 + ... + n, and 3 is the second of them.¹^•⁶ Fermat's polygonal number theorem states that every positive integer is a sum of at most three triangular numbers; Carl Friedrich Gauss proved this triangular case and recorded it in his diary on July 10, 1796 with the notation "ΕΥΡΗΚΑ! num = Δ + Δ + Δ".⁶ The number 3 itself has two such representations, {3, 0, 0} and {1, 1, 1}.⁷

Historical and cultural notes

The digit 3 reached its modern European form through the Hindu-Arabic numerals, whose transmission from India through the Arabic world and into Europe took centuries; the numerals appear in the Codex Vigilanus, copied by a monk in Spain in 976, and were popularized in Europe by Fibonacci's Liber abaci of 1202.⁸ In the eastern Arabic empire the numerals were known as Gubar ("dust") numerals, from the dust board used for calculation.⁸

The number three carries heavy symbolic weight across traditions. In ancient Egyptian temple theology, divine triads such as Osiris, Isis, and Horus organized local religion, and the triangle represented harmony and balance in Egyptian mythology.⁹^•¹⁰ In Christian doctrine, the Trinity defines one God existing in three co-equal, coeternal, and consubstantial divine persons: God the Father, God the Son, and God the Holy Spirit.¹⁰

References

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Special and named integers › Named individual integers

Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026

Notice something wrong?

3