Draft:−3

  • 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)


← −4 −3 −2 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinalnegative three
Ordinal-3rd
(negative third)
Divisors1, 3
Arabic−٣
Bengali−৩
Binary (byte)11111101

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 , negative three is its fundamental discriminant, and the ring of integers in this field is the lattice of Eisenstein integers.[7]

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 , 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.

See Also

References

  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.
  7. ^ a b c "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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