{ "id": "ep3424rtss", "slug": "3", "title": "3", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.number", "label": "Numbers and algebra", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number" }, { "id": "physical.math.number.arithmetic-number-systems", "label": "Arithmetic and number systems", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number.arithmetic-number-systems" }, { "id": "physical.math.number.integer-sequences-and-partitions", "label": "Integer sequences and partitions", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number.integer-sequences-and-partitions" }, { "id": "physical.math.number.special-named-integers", "label": "Special and named integers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number.special-named-integers" }, { "id": "physical.math.number.special-named-integers.named-individual-integers", "label": "Named individual integers", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number.special-named-integers.named-individual-integers" } ], "geo": [ { "id": "geo.nongeo.nodate.physical.math.number.integer-sequences-and-partitions", "label": "Non-geographic · Not dated: Integer sequences and partitions", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number.integer-sequences-and-partitions", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.nodate", "label": "Non-geographic · Not dated", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate" }, { "id": "geo.nongeo.nodate.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical" }, { "id": "geo.nongeo.nodate.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math" }, { "id": "geo.nongeo.nodate.physical.math.number", "label": "Numbers and algebra", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number" }, { "id": "geo.nongeo.nodate.physical.math.number.arithmetic-number-systems", "label": "Arithmetic and number systems", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number.arithmetic-number-systems" }, { "id": "geo.nongeo.nodate.physical.math.number.integer-sequences-and-partitions", "label": "Integer sequences and partitions", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number.integer-sequences-and-partitions" } ] } ], "excerpt": "3 (three) is the smallest odd prime number, the only prime one more than another prime (2), and a member of the Fibonacci and triangular number sequences.", "snippet": "3 (three) is the smallest odd prime number, the only prime one more than another prime (2), and a member of the Fibonacci and triangular number sequences.", "node": "physical.math.number.special-named-integers.named-individual-integers", "markdown": "# 3\n\n**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](https://oeis.org/wiki/3) 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](https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2022/12/amstext35.pdf)\n\n| Key fact | Detail |\n|---|---|\n| Primality | Smallest odd prime; the only prime one more than another prime (2)[1](https://oeis.org/wiki/3) |\n| Consecutive primes | 2 and 3 are the only consecutive primes; every other pair of primes differs by at least 2[2](https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2022/12/amstext35.pdf) |\n| Divisibility test | An integer is divisible by 3 if and only if its digit sum is[3](https://www.open.edu/openlearn/ocw/pluginfile.php/1044022/mod_resource/content/1/17mst125u3-extract.pdf) |\n| Sequence membership | Appears among the odd numbers, primes, Fibonacci numbers, Lucas numbers, and triangular numbers[1](https://oeis.org/wiki/3) |\n| Partitions | Exactly three partitions of 3: {1, 1, 1}, {1, 2}, and {3}[1](https://oeis.org/wiki/3) |\n| Approximate values | \\( \\sqrt{3} \\approx 1.73205080 \\); \\( \\log_{2} 3 \\approx 1.58496250 \\)[1](https://oeis.org/wiki/3) |\n| Representations | 11 in binary, 10 in ternary, 111 in negabinary[1](https://oeis.org/wiki/3) |\n\n## Arithmetic character\n\nAs 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](https://oeis.org/wiki/3) • [4](https://mathworld.wolfram.com/PrimeNumber.html) 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](https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2022/12/amstext35.pdf) 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](https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2022/12/amstext35.pdf)\n\nThe number 3 also appears in the classical sequences: it is the third odd number, the second prime, the fourth [Fibonacci](https://www.edgepedia.org/fibonacci) number, the third [Lucas number](https://www.edgepedia.org/lucas-number), and the second triangular number.[1](https://oeis.org/wiki/3) In Pascal's triangle it occurs twice, and in Lozanić's triangle four times.[1](https://oeis.org/wiki/3)\n\n## The digit-sum divisibility test\n\nThe 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](https://www.open.edu/openlearn/ocw/pluginfile.php/1044022/mod_resource/content/1/17mst125u3-extract.pdf) 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\n\\[ n = \\sum_{i=0}^{k} d_{i} \\cdot 10^{i} \\equiv \\sum_{i=0}^{k} d_{i} \\pmod{3}. \\]\n\nThe 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](https://www.open.edu/openlearn/ocw/pluginfile.php/1044022/mod_resource/content/1/17mst125u3-extract.pdf) The rule is a special case of modular arithmetic, the system of congruences published by [Carl Friedrich Gauss](https://www.edgepedia.org/carl-friedrich-gauss) in 1801, in which two integers are congruent modulo n when they leave the same remainder on division by n.[5](https://www.sw.it.aoyama.ac.jp/2024/Math1/lecture13.html)\n\n## Powers, roots, and logarithms\n\nThe powers of 3 begin 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, forming the sequence of powers of three.[1](https://oeis.org/wiki/3) 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](https://oeis.org/wiki/3)\n\n## Behavior in other number systems\n\nIn 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](https://www.edgepedia.org/negabinary), since \\( (-2)^{2} + (-2)^{1} + (-2)^{0} = 4 - 2 + 1 = 3 \\).[1](https://oeis.org/wiki/3)\n\nIn 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](https://oeis.org/wiki/3) 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](https://oeis.org/wiki/3)\n\nPrimality 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](https://oeis.org/wiki/3)\n\n## Sums of three triangular numbers\n\nThe triangular numbers 1, 3, 6, 10, 15, ... are the sums 1 + 2 + ... + n, and 3 is the second of them.[1](https://oeis.org/wiki/3) • [6](https://mathworld.wolfram.com/TriangularNumber.html) 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](https://mathworld.wolfram.com/TriangularNumber.html) The number 3 itself has two such representations, {3, 0, 0} and {1, 1, 1}.[7](https://oeis.org/wiki/Triangular_numbers)\n\n## Historical and cultural notes\n\nThe digit 3 reached its modern European form through the Hindu-[Arabic numerals](https://www.edgepedia.org/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](https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_numerals/) In the eastern Arabic empire the numerals were known as Gubar (\"dust\") numerals, from the dust board used for calculation.[8](https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_numerals/)\n\nThe number three carries heavy symbolic weight across traditions. In ancient [Egyptian temple](https://www.edgepedia.org/egyptian-temple) theology, divine triads such as Osiris, Isis, and Horus organized local religion, and the triangle represented harmony and balance in [Egyptian mythology](https://www.edgepedia.org/egyptian-mythology).[9](https://codexnumerica.com/number/3.html) • [10](https://reference-global.com/download/article/10.2478/zireb-2023-0010.pdf) In Christian doctrine, the Trinity defines one God existing in three co-equal, coeternal, and consubstantial divine persons: [God the Father](https://www.edgepedia.org/god-the-father), God the Son, and God the Holy Spirit.[10](https://reference-global.com/download/article/10.2478/zireb-2023-0010.pdf)\n\n## References\n\n1. [3, OeisWiki](https://oeis.org/wiki/3)\n2. [The Prime Numbers (chapter from an AMS preliminary textbook, L. Alloza et al.)](https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2022/12/amstext35.pdf)\n3. [Divisibility tests, MST124 Book 3 Unit 17 extract, The Open University](https://www.open.edu/openlearn/ocw/pluginfile.php/1044022/mod_resource/content/1/17mst125u3-extract.pdf)\n4. [Prime Number, Wolfram MathWorld](https://mathworld.wolfram.com/PrimeNumber.html)\n5. [Modular Arithmetic, Discrete Mathematics I, Martin J. Dürst, Aoyama Gakuin University](https://www.sw.it.aoyama.ac.jp/2024/Math1/lecture13.html)\n6. [Triangular Number, Wolfram MathWorld](https://mathworld.wolfram.com/TriangularNumber.html)\n7. [Triangular numbers, OeisWiki](https://oeis.org/wiki/Triangular_numbers)\n8. [The Arabic numeral system, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Arabic_numerals/)\n9. [The Number 3 in Sacred Traditions, Codex Numerica](https://codexnumerica.com/number/3.html)\n10. [Numeral 3 in symbolism and religion, Zeitschrift für Interkulturelle Erziehungswissenschaft (2023)](https://reference-global.com/download/article/10.2478/zireb-2023-0010.pdf)\n\n---\n*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*\n\n*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/3", "markdown_url": "https://www.edgepedia.org/3.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"3\", Edgepedia (EdgeChat), https://www.edgepedia.org/3. Edgepedia Community License 1.0.", "credit_md": "\"[3](https://www.edgepedia.org/3)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/3](https://www.edgepedia.org/3). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"3\", Edgepedia (EdgeChat), https://www.edgepedia.org/3. Edgepedia Community License 1.0.", "speakable": "3 is the smallest odd prime number, the only prime one more than another prime, and a member of the Fibonacci and triangular number sequences." }