Draft:−3

← −4 −3 −2 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers; Integers;
Cardinal	negative three
Ordinal	-3rd; (negative third)
Divisors	1, 3
Arabic	−٣
Bengali	−৩
Binary (byte)	11111101

Submission declined on 5 July 2026 by Stuartyeates (talk).

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Declined by Stuartyeates 23 days ago. Last edited by Stuartyeates 23 days ago. Reviewer: Inform author.

Resubmit

Please note that if the issues are not fixed, the draft will be declined again.

Submission declined on 19 June 2026 by Stuartyeates (talk).

As before. You need to read and act on the feedback.

Declined by Stuartyeates 40 days ago.

Submission declined on 13 June 2026 by Stuartyeates (talk).

This draft reads like an essay or opinion piece. Wikipedia is not a place for original research or personal opinions. The draft should:

summarize secondary sources: do not offer your own analysis or arguments;
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Instead, only summarize in your own words a range of independent, reliable, published sources that discuss the subject.

Declined by Stuartyeates 45 days ago.

Submission declined on 12 June 2026 by Hurricane Wind and Fire (talk).

This draft appears to be a dictionary definition. Wikipedia is not a dictionary and we do not accept articles that are simply definitions of words, acronyms, or slang.

To improve the draft it must expand on the subject, such as discussing its history, cultural significance, or impact rather than just defining the word;
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Declined by Hurricane Wind and Fire 47 days ago.

Submission declined on 26 February 2026 by Spiderone (talk).

This draft is not adequately supported by reliable sources. Wikipedia's verifiability policy requires that all content be supported by reliable sources.

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This draft appears to be a dictionary definition. Wikipedia is not a dictionary and we do not accept articles that are simply definitions of words, acronyms, or slang.

To improve the draft it must expand on the subject, such as discussing its history, cultural significance, or impact rather than just defining the word;
To create a dictionary entry you can contribute to our sister project Wiktionary. Please ensure the subject meets Wiktionary's inclusion criteria.

Declined by Spiderone 5 months ago.

Submission declined on 1 November 2025 by GGOTCC (talk).

This draft appears to be a dictionary definition. Wikipedia is not a dictionary and we do not accept articles that are simply definitions of words, acronyms, or slang.

To improve the draft it must expand on the subject, such as discussing its history, cultural significance, or impact rather than just defining the word;
To create a dictionary entry you can contribute to our sister project Wiktionary. Please ensure the subject meets Wiktionary's inclusion criteria.

Declined by GGOTCC 8 months ago.

Comment: Per comment by User:Hurricane Wind and Fire, needs a properties section on a par with -2. Stuartyeates (talk) 23:15, 13 June 2026 (UTC)

Comment: Still largely just a dictionary definition. Expand onto the properties section to a point where it is similar to the properties section in -2 and it will be ready for acceptance. 🌀Hurricane Wind and Fire, why did you decline my draft? (talk) (contribs)🔥 03:41, 12 June 2026 (UTC)

Comment: If you want the draft to be like -2, then please include citations so this is not a dictionary entry GGOTCC 21:07, 1 November 2025 (UTC)

In mathematics, negative three or minus three is an negative integer three units from the origin, denoted as −3 or ⁻3. It is the additive inverse of 3, positioned between −4 and −2. It is the second largest negative odd number.

Properties

Negative three is a quadratic residue modulo 4.^[1]
Negative three is the first, and smallest negated heegner number. only nine negative numbers have this property: −3, −4, −7, −8, −11, −19, −43, −67, −163.^[2]
Negative three is the only negative integer whose Möbius function value appears as a fixed point of the map in the sequence 𝜇(𝑛) ∈ {−1,0,1}.^[3]

Divisors of negative three

The divisors of the number negative three, including negative divisors, are identical to those of two: 1, 3, −1 −3.^[4] Since its only divisors are ±1 and ±3, negative three is considered an irreducible element, which is the equivalent of a prime number for negative integers.^[5]

Square root of negative three

The square root of negative three produces both quadratic field, and cyclotomic field.^[6]

In the quadratic field $\mathbb {Q} ({\sqrt {-3}})$ , negative three is its fundamental discriminant, and the ring of integers in this field is the lattice of Eisenstein integers.^[7]

{\mathit {Z}}[\omega ],\omega ={\frac {-1+{\sqrt {-3}}}{2}}

which forms a perfect hexagonal (triangular) lattice in the plane.^[7]

The quadratic field also has class number 1, meaning it behaves better as the integers for factoring.^[8]

In the cyclotomic field $\mathbb {Q} (\zeta _{3})$ , the discriminant of the field is −3,^[7] meaning three is the only rational prime that becomes ramified,^[9]^[10] and the ramification index is exactly two.

References

^ Burton, David (February 15, 2006). Elementary Number Theory. Waveland Press. Retrieved 11 June 2026.
^ Sloane, N. J. A. (ed.). "Sequence A014602 (Discriminants of imaginary quadratic fields with class number 1 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A008683". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Sloane, N. J. A. (ed.). "Sequence A027750 (Triangle read by rows in which row n lists the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ "Unique Factorization Domains (UFDs)" (PDF). University of Galway. Retrieved 19 June 2026.
^ "Cyclotomic Field". Wolfram MathWorld. Retrieved 27 June 2026.
^ ^a ^b ^c "Eisenstein Integer". Wolfram MathWorld. Retrieved 12 June 2026.
^ Sloane, N. J. A. (ed.). "Sequence A000924 (Class number of Q(sqrt(-n)), n squarefree.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ "Quadratic Field". Wolfram MathWorld. Retrieved 27 June 2026.
^ Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Springer Science & Business Media. ISBN 978-0-387-94762-4.

This article about a number is a stub. You can help Wikipedia by adding missing information.

[1] Burton, David (February 15, 2006). Elementary Number Theory. Waveland Press. Retrieved 11 June 2026.

[2] Sloane, N. J. A. (ed.). "Sequence A014602 (Discriminants of imaginary quadratic fields with class number 1 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[3] Sloane, N. J. A. (ed.). "Sequence A008683". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[4] Sloane, N. J. A. (ed.). "Sequence A027750 (Triangle read by rows in which row n lists the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[5] "Unique Factorization Domains (UFDs)" (PDF). University of Galway. Retrieved 19 June 2026.

[6] "Cyclotomic Field". Wolfram MathWorld. Retrieved 27 June 2026.

[:0-7] "Eisenstein Integer". Wolfram MathWorld. Retrieved 12 June 2026.

[8] Sloane, N. J. A. (ed.). "Sequence A000924 (Class number of Q(sqrt(-n)), n squarefree.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[9] "Quadratic Field". Wolfram MathWorld. Retrieved 27 June 2026.

[10] Washington, Lawrence C. (1997). Introduction to Cyclotomic Fields. Springer Science & Business Media. ISBN 978-0-387-94762-4.

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This page was redesigned at the edge.

Draft:−3

Properties

Divisors of negative three

Square root of negative three

See Also

References