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).

This draft's references do not show that the subject meets Wikipedia's criteria for inclusion. The draft requires multiple published secondary sources that:

provide significant coverage: discuss the subject in detail, not just brief mentions or routine announcements;
are reliable: from reputable outlets with editorial oversight;
are independent: not connected to the subject, such as interviews, press releases, the subject's own website, or sponsored content.

Please add references that meet all three of these criteria. If none exist, the subject is not yet suitable for Wikipedia.

If you would like to continue working on the submission, click on the "Edit" tab at the top of the window.
If you have not resolved the issues listed above, your draft will be declined again and potentially deleted.
If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors.
Please do not remove reviewer comments or this notice until the submission is accepted.

Where to get help

If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews.
If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed.

How to improve a draft

Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia.
Help:Wikitext – how to use the markup
Help:Referencing for beginners – how to include references
Wikipedia:Article development – how to develop your article
Wikipedia:Writing better articles – how to improve your article
Wikipedia:Verifiability – make sure your article includes reliable third-party sources

You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article.

Improving your odds of a speedy review

To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags.

Add tags to your draft

Editor resources

Easy tools: Citation bot (help) | Advanced: Fix bare URLs

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;
be written from a neutral point of view: represent the subject without bias, avoiding praise, criticism, or persuasive or promotional language;
not contain original research: do not include new theories, unpublished ideas, or personal experiences.

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;
To create a dictionary entry you can contribute to our sister project Wiktionary. Please ensure the subject meets Wiktionary's inclusion criteria.

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.

Reliable sources include: reputable newspapers, magazines, academic journals, and books from respected publishers.
Unacceptable sources include: personal blogs, social media, predatory publishers, most tabloids, and websites where anyone can contribute.

Replace any unreliable sources with high-quality sources. If you cannot find a reliable source for the material, it should be removed.

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.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

This page was redesigned at the edge.

A page that knows where it met you.

Draft:−3

Properties

Divisors of negative three

Square root of negative three

See Also

References