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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 can be written as \(x^2+xy+y^2\) for integers \(x\) and \(y\).[1] The first values are \(0,1,3,4,7,9,12,13,16\).[1] The defining object is the set of values of this fixed quadratic form, rather than a particular pair of coordinates or an application that happens to use the form.[1]

Choose two unit vectors at an angle of 60 degrees on a triangular lattice. The squared length of the vector with integer coordinates \((x,y)\) is \(x^2+xy+y^2\), by the cosine rule.[1][2] Equivalently, a suitable choice of coordinates identifies the same value with the algebraic norm of an Eisenstein integer.[2] These interpretations make the arithmetic family visible as lattice geometry; the stated unit length matters when comparing the numerical squared length with a differently normalized root lattice.[2]

For a positive integer, there is a second, exact test: it belongs to the family if and only if every prime factor congruent to \(2\pmod 3\) occurs to an even power.[3] Zero is included directly by \((x,y)=(0,0)\), outside that prime-factor statement.[1] Thus \(7=2^2+2\cdot1+1^2\) qualifies, whereas \(2\) does not.

Structural Signature

  • Integral coordinates. A pair \((x,y)\) of integers selects a lattice vector; allowing arbitrary real coordinates would erase the discrete membership boundary.
  • Fixed quadratic form. The operation \(Q(x,y)=x^2+xy+y^2\) assigns a nonnegative integer to that pair. Its cross term is part of the identity, not a replaceable decoration.
  • Represented value. The output \(n=Q(x,y)\) is a member even when more than one coordinate pair gives \(n\). The number family is the image of the form, not a list of coordinate pairs.
  • Equivalent arithmetic test. For \(n>0\), the even-exponent condition on primes \(2\pmod 3\) decides the same membership question without constructing a coordinate pair. It characterizes the family but does not add a second defining requirement.[3]
  • Boundary. An integer such as \(2\) or \(5\) fails because its factorization has a \(2\pmod 3\) prime to an odd power; a noninteger length or a count from an unrelated model is not admitted by its resemblance to a hexagon.

What It Is Not

  • Not every nonnegative integer. The discrete lattice rule excludes \(2\) and \(5\); an unrestricted count has no reason to satisfy it.[3]
  • Not merely a square number. Setting \(y=0\) yields every square, but \(3=1^2+1\cdot1+1^2\) and \(7=2^2+2\cdot1+1^2\) are nonsquare members.
  • Not an Eisenstein integer itself. An Eisenstein integer is an algebraic element with coordinates and multiplication; a Löschian number is an ordinary nonnegative integer value of its norm. Different elements can have the same norm.
  • Not a physical theory of every hexagonal object. The cellular and capsid models below use the form under their own geometric assumptions. The number-theoretic membership rule does not predict radio interference, protein count, or biological feasibility by itself.[4][5]

Scope of Application

In number theory the family is the set represented by \(Q\), with the prime-factor criterion as an exact positive-integer membership test.[3] In unit-edge triangular-lattice geometry it is the set of squared lengths from the origin to lattice points. These are two descriptions of one mathematical carrier, not two different definitions.[1]

The same numerical form also appears in idealized hexagonal cellular frequency-reuse clusters and in triangular-lattice descriptions of icosahedral viral capsids. In the first, it indexes possible reuse-cluster sizes; in the second, a triangulation number. They supply unlike applications of the mathematical family, with unlike physical constraints.[4][5]

Clarity

The name separates three questions often compressed into one formula. Is an integer represented? The prime condition gives a yes-or-no answer for positive values. Which coordinates represent it? A witness pair supplies geometry and can carry direction or handedness information that the bare value loses. What does the value mean in a model? A cellular \(N\) and a capsid \(T\) may both equal seven while indexing different objects. Stating the question prevents a membership result from being mistaken for a full physical explanation.[3][4][5]

Manages Complexity

The represented-value rule compresses infinitely many coordinate pairs into one recognizable integer family. The prime-factor theorem compresses a potentially open-ended coordinate search into a finite factorization check. For example, the factorization of \(14=2\cdot7\) rules it out because \(2\) occurs to an odd power; \(28=2^2\cdot7\) passes the arithmetic test.[3]

This compression drops information. A value alone does not specify a vector orientation or the particular lattice construction. In a capsid model, interchanging \(h\) and \(k\) can preserve \(T\) while changing the geometric handedness; the number-theoretic set cannot carry that distinction.[5]

Abstract Reasoning

To classify \(n>0\), factor it, inspect the exponents of every prime congruent to \(2\pmod3\), and conclude membership exactly when each such exponent is even. Test \(n=0\) by the defining coordinates instead. A passing factorization proves that some integer pair represents \(n\); it does not select a pair for a geometric construction.[3]

The rule also licenses bounded deductions. Every integer square is represented by \((x,0)\); infinitely many distinct squares therefore lie in the family.[1] More generally, substituting \(y=mx\) yields \(Q(x,mx)=(m^2+m+1)x^2\), another directly derived subfamily.[1] If two positive members are multiplied, the restricted prime exponents remain even, so their product is a member.[3][1] The same exponent test shows that the greatest common divisor and least common multiple of positive members remain members, because the minimum or maximum of even exponents is even.[3] Since \(1\) belongs, this multiplicative closure also makes the family a monoid.[1]

The form is even exactly when both coordinates are even: checking the four parity pairs shows that \(00\) yields even and \(01,10,11\) yield odd.[1] This does not establish a 75/25 probability among distinct represented numbers, because multiple coordinate pairs can produce the same value. Without a specified sampling rule and a theorem for distinct values, no numerical probability follows from this parity calculation.

Zero needs separate handling: it is a member, and \(0\) times an integer nonmember is again member \(0\).[1] Thus the unqualified claim that a member times a nonmember is never a member is false. For a positive member times a positive integer nonmember, a restricted odd exponent remains odd; for a negative integer nonmember, the product is negative and therefore outside the family.[3]

Knowledge Transfer

Within number theory, coordinate representation and prime-factor testing can be used interchangeably for the membership verdict. The coordinate view is useful when the question asks for a lattice witness; the factor view is useful when it asks only whether the integer is possible. The same form transfers literally to hexagonal model indices: \(N\) and \(T\) can both be checked for representability, even though their interpretations diverge.[4][5]

The transfer stops at the arithmetic test. A possible cellular cluster size does not establish a good signal-to-interference ratio, and a possible triangulation number does not ensure a particular virus makes that shell. Those conclusions need the separate model assumptions and observations in each field.[4][5]

Examples

Canonical: a seven-cell reuse cluster

Zhang illustrates a seven-cell frequency-reuse pattern in an ideal hexagonal cellular geometry and states that a cluster size has the form \(N=i^2+ij+j^2\).[4] Substituting \((i,j)=(2,1)\) gives \(N=4+2+1=7\); the coordinate pair is this article's calculation from the source's formula and diagram, not a value printed beside that diagram. Mapped back: integral coordinates \((2,1)\) → fixed form \(Q\) → represented value \(7\) → valid index for that idealized reuse cluster.[4] The radio meaning of \(N\) is additional to the arithmetic.

Applied: a viral capsid triangulation number

Luque and Reguera's original geometric study presents a \(T=3\) lattice construction at \((h,k)=(1,1)\), for which \(Q=1+1+1=3\), and illustrates the \(T=7l\) geometry of bacteriophage HK97 at \((2,1)\).[5] Mapped back: integral lattice steps \((1,1)\) or \((2,1)\) → the same fixed form \(Q\) → represented triangulation value \(3\) or \(7\).[5] The capsid interpretation adds geometry, handedness, and biological constraints absent from the integer's definition.

Structural Tensions

T1: A fast membership test versus a constructive witness. Factorization decides whether a positive integer is in the family without exhibiting coordinates; a coordinate pair shows a direction and construction but may take more work to find. Neither output answers every question. Diagnostic: if the decision is merely admissibility, use the prime-exponent test; if a reuse layout or capsid facet must be drawn, require coordinates and the destination model.[3]

T2: Shared arithmetic versus separate physical meanings. Treating cellular \(N\) and capsid \(T\) as the same represented-value family carries a valid arithmetic check across settings. Treating them as the same physical mechanism would import unsupported radio or protein consequences; keeping them wholly separate would hide their common representability restriction. Diagnostic: transfer \(Q\) and its membership result, then ask anew what the target model says the index counts.[4][5]

Structural–Framed Character

The entry is structural within mathematics and domain framed as a named number family. Its generating rule and membership theorem are exact, while use of the numbers as indices in radio or capsid models depends on those models.

  • Evaluative weight: membership is descriptive. Seven is not a better or worse number because it belongs; suitability of a seven-cell radio design is a separate engineering judgment.

  • Human-practice dependence: the form defines a mathematical set independent of a laboratory or institution. Choosing unit scale, notation, and an applied model is a human practice that must be stated when interpreting a value.

  • Institutional origin: the name honors August Lösch, but the criterion does not require his market theory or a particular institution.[1]

  • Vocabulary travel: quadratic form, lattice norm, and factorization travel through mathematics; the name “Löschian” continues to pick out this one represented-value set, even when another field uses the same formula.
  • Import versus recognition: one recognizes a member by the fixed form or equivalent prime rule. An application imports that arithmetic into a model; neither a label nor a matching count proves the model's physical assumptions.

Its character: a precise formal class with limited, model-dependent applications beyond number theory.

Structural Core vs. Domain Accent

The possible portable skeleton is “values generated by a fixed rule and tested by an equivalent criterion.” Its wider status is an explicit future-Prime question: no live Prime passed the full-signature typed-parent test for this named set. Here the rule is specifically \(x^2+xy+y^2\) on integer pairs, with Eisenstein/triangular geometry and the restricted-prime membership theorem. Removing those commitments leaves only generic representability. The named entry therefore remains domain specific and is a reviewed unparented root in the official graph.

Factorization is useful for applying the positive-member theorem, and Representation names the broader act of expressing a value in a form. Neither term defines the Löschian family or passes the strict typed-parent test for this set. Classification describes the act of sorting integers by a rule, not the set that results. The DAG therefore asserts no edge and marks a reviewed unparented root.

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 Number is a proper subset because \(Q(x,0)=x^2\), while \(3\) and \(7\) are Löschian but nonsquare. Quadratic Integer concerns an element of a quadratic field; a Löschian number is a norm value, so element and output must not be merged. The same quadratic expression in an applied model is a qualifying Löschian value only as arithmetic; the model's physical conclusions require separate evidence.

References

[1] OEIS Foundation, “A003136: Loeschian numbers,” definition, terms, Comments, and Formula, https://oeis.org/A003136. Authoritative curated sequence; used for the number family, included zero, lattice interpretation, and multiplicative property, not for an unsourced odd/even probability. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[2] Eric W. Weisstein, “Löschian Number,” MathWorld, https://mathworld.wolfram.com/LoeschianNumber.html. Authoritative mathematical reference for the Eisenstein-norm interpretation and terminology; geometric unit-basis identity is an explicit cosine-rule derivation. 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 mathematical paper for the positive-integer prime-factor criterion; zero is handled by direct substitution. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k

[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-hosted textbook excerpt; directly supplies the seven-cell illustration and \(N=i^2+ij+j^2\), while \((2,1)\) is calculated here. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h

[5] Antoni Luque and David Reguera, “The Structure of Elongated Viral Capsids,” Biophysical Journal 98, no. 12 (2010): 2993–3003, Fig. 1 and introductory Caspar–Klug construction, https://doi.org/10.1016/j.bpj.2010.02.051. Original research article; the worked \(T=3\) and HK97 \(T=7l\) mappings belong to its geometric model, not to a universal biological rule. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i