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Löschian Number

A nonnegative integer represented by the quadratic form x² + xy + y² with integer coordinates.

Version
v1 · 2026-10-25 · History
Domain-specific #
14411
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Number Theory, Quadratic Forms → Mathematics
Aliases
Loeschian number

Core Idea

A Löschian number is a nonnegative integer that equals \(x^2+xy+y^2\) for some integers \(x,y\).[1] Examples start \(0,1,3,4,7,9,12,13,16\).[1] Geometrically, choose two unit directions at 60 degrees on a triangular lattice: by the cosine rule this expression is the squared length reached by integer steps along them.[1][2] It is also an Eisenstein-integer norm under an equivalent coordinate choice.[2]

For a positive integer there is an exact shortcut: it is Löschian precisely when every prime factor congruent to \(2\pmod3\) has an even exponent.[3] Zero belongs because \(Q(0,0)=0\), without needing a prime factorization.

Scope of Application

This is a number-theoretic family. The same quadratic form appears as a possible cluster-size index \(N\) in an ideal hexagonal cellular-frequency-reuse model and as a triangulation index \(T\) for modeled icosahedral virus capsids.[4][5] Those models give different meanings to the number; the arithmetic test alone does not settle radio performance or capsid biology.

Clarity

Keep membership, coordinates, and application apart. The factorization rule can answer whether a positive integer belongs; a pair \((x,y)\) shows a particular lattice construction; a field model explains what its index counts. A shared value such as seven does not make a cellular cluster and a viral shell the same object.

Manages Complexity

The prime rule can rule out a value without searching coordinate pairs. For instance \(14=2\cdot7\) is excluded because the \(2\) has odd exponent, while \(28=2^2\cdot7\) passes.[3] The answer is about representability, not a chosen vector direction.

Abstract Reasoning

Factor a positive proposed value, check the exponents of primes \(2\pmod3\), and conclude membership only if all are even.[3] Every square belongs by taking \(y=0\), so there are infinitely many members.[1] Multiplication of members preserves membership; \(1\) is an identity and \(0\) is an absorbing member.[1] The zero case matters: \(0\) times an integer nonmember is member \(0\), so an unconditional “member times nonmember is never a member” rule is false.[1]

Knowledge Transfer

The same integer form can be checked in cellular and capsid models, but its physical implications do not transfer. Use \(x^2+xy+y^2\) to test the numerical index, then use each field's own evidence for interference, geometry, or biological realization.[4][5]

Examples

  • Seven-cell reuse. Zhang shows a seven-cell cluster in ideal hexagonal geometry with \(N=i^2+ij+j^2\).[4] Substituting \((i,j)=(2,1)\) gives \(N=7\); the pair is a calculation from the printed formula. The integer coordinates, form, and output fill the three defining roles.[4]

  • Capsid triangulation. Luque and Reguera show \(T=3\) with \((h,k)=(1,1)\) and the \(T=7l\) HK97 construction with \((2,1)\).[5] The same coordinate rule produces the value, while the capsid model supplies a different interpretation.[5]

Neighborhood in Abstraction Space

Löschian Number sits in a sparse region of the domain-specific corpus (86th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2769 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With

  • Square numbers are members, but not all members are squares: \(3=Q(1,1)\) and \(7=Q(2,1)\).[1]

  • A quadratic integer is an algebraic element; a Löschian number is a norm value.[2]

  • An arbitrary positive count is not Löschian: \(2\) and \(5\) fail the prime-factor test.[3]

References

[1] OEIS Foundation, “A003136: Loeschian numbers,” definition, terms, Comments, and Formula, https://oeis.org/A003136. Authoritative curated sequence for the family, zero, and multiplicative property. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g

[2] Eric W. Weisstein, “Löschian Number,” MathWorld, https://mathworld.wolfram.com/LoeschianNumber.html. Authoritative mathematical reference for the Eisenstein norm and lattice interpretation; the unit-basis squared-length identity follows directly by the cosine rule. registry ↩a ↩b ↩c

[3] Umesh P. Nair, “Elementary Results on the Binary Quadratic Form \(a^2+ab+b^2\)” (2004), Theorem 16 and proof, PDF pp. 6–7, https://arxiv.org/pdf/math/0408107. Original proof of the positive-integer prime-factor criterion. registry ↩a ↩b ↩c ↩d

[4] Keith Q. T. Zhang, Wireless Communications: Principles, Theory and Methodology (Wiley, 2016), chapter 1 §1.8, printed p. 9 and Fig. 1.2, https://catalogimages.wiley.com/images/db/pdf/9781119978671.excerpt.pdf. Publisher excerpt for the ideal hexagonal \(N=7\) illustration and formula. registry ↩a ↩b ↩c ↩d

[5] Antoni Luque and David Reguera, “The Structure of Elongated Viral Capsids,” Biophysical Journal 98, no. 12 (2010): 2993–3003, Fig. 1, https://doi.org/10.1016/j.bpj.2010.02.051. Original research article for the capsid triangulation examples. registry ↩a ↩b ↩c ↩d