The Number of Idoneal Numbers

Completeness of the List of Idoneal Numbers

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

An idoneal positive integer $n$ can be characterized by the proper ideal class group of the imaginary quadratic order of discriminant $-4n$: every ideal class has order dividing two. Equivalently one uses proper classes of primitive positive definite binary quadratic forms of that discriminant. The target is the completeness of the classical list, not merely a search up to a fixed bound. [S1] [S2]

Statement recorded in the supplied source.

Are there exactly 65 idoneal numbers, or could there be 66 or 67? [S1]

Residual task reported by the archive.

The embedded review $2026-08-17$ records: Exclude or exhibit the one or two possible exceptional idoneal numbers without assuming GRH. [S1]

Short English statement

Is the classical collection of sixty-five idoneal integers complete, or do further examples exist?

Research attempt

Scope and approach.

The class group in the definition belongs to the order of discriminant $-4n$, which need not be the maximal order of its quadratic field. Replacing it by the maximal-order class group could change the answer. Kani's primary survey [E1] treats this distinction and corrections to historical formulations. This attempt uses the stated order throughout and constructs an exact finite test by reduced forms. It does not assume the generalized Riemann hypothesis or an effective global cutoff.

A complete finite test for a fixed discriminant.

Write a primitive positive definite form as $(a,b,c)$ with $b^2-4ac=-4n$. Use the usual reduced representative convention \[ |b|\le a\le c,\quad \gcd(a,b,c)=1, \quad b\ge0\text{ if }|b|=a\text{ or }a=c. \] The reduction theorem for positive definite forms gives one such representative per proper class under this boundary convention. This standard theorem is an input from quadratic-form theory; it is not reproved here. The bound needed for enumeration follows immediately: \[ 4n=4ac-b^2\ge4a^2-a^2=3a^2, \qquad a\le\sqrt{4n/3}. \] Thus enumerate those finitely many positive $a$, the integers $-a\le b\le a$, and retain $c=(b^2+4n)/(4a)$ when integral and satisfying the displayed conditions. This proves coverage at fixed $n$.

Detecting exponent two without a composition table.

The inverse of the proper class represented by $(a,b,c)$ is represented by $(a,-b,c)$. In the interior $0<|b|<a<c$, these are distinct reduced representatives, so the class cannot equal its inverse. On the boundary, the cases $b=0$, $|b|=a$, or $a=c$ are exactly the ambiguous reduced forms, whose proper classes equal their inverses. For example the integral substitutions changing a variable by the other, or interchanging the two variables with a sign, identify the boundary inverse with its chosen representative. These substitutions have determinant one. Consequently the required class-group exponent condition is equivalent to every enumerated form satisfying at least one of \[ b=0,\qquad |b|=a,\qquad a=c. \] The test uses proper classes and their inverses, not improper equivalence, which would identify inverse pairs and lose the obstruction.

Small complete examples.

For $n=5$ the reduced forms are $(1,0,5)$ and $(2,2,3)$; both are ambiguous. For $n=6$ they are $(1,0,6)$ and $(2,0,3)$, again both ambiguous. For $n=11$ there are three forms \[ (1,0,11),\quad (3,-2,4),\quad(3,2,4). \] The latter two are distinct inverse classes in the strict interior, so eleven is not idoneal under the catalogue criterion. These examples illustrate an actual obstruction, rather than merely checking that a class number is a power of two. A group of order a power of two may contain elements of order four or higher, so that numerical class-number condition alone is insufficient.

What finite reduction does and does not bound.

The inequality $a\le\sqrt{4n/3}$ bounds representatives after $n$ has been fixed; it puts no bound on $n$. To prove completeness of the classical list by this route one needs to show that every larger discriminant has a primitive interior reduced form with nonzero $b$, or obtain a separate effective discriminant bound. An algorithm which always halts for each individual $n$ can still run forever over all $n$ when no further idoneal value exists. Thus a decision procedure at fixed input is different from a proof that no exceptional input remains.

One possible direct obstruction is to exhibit $a,b$ with $0<|b|<a<c=(b^2+4n)/(4a)$ and $\gcd(a,b,c)=1$. The divisibility $4a\mid b^2+4n$ is a local condition and primitivity is another. Producing such witnesses uniformly in all sufficiently large $n$ is precisely the unproved step of this elementary approach. Selecting $a$ near $\sqrt n$ by an approximation does not ensure this congruence, and discarding nonmaximal orders does not preserve the target.

Executed finite audit.

The integer script enumerates all primitive reduced forms for every $1\le n\le5000$ and applies the inverse-class criterion. Exactly $65$ values pass; the largest is $1848$. The full list and sample form sets, including all sixteen forms for $n=1848$, are in results_second.json. The code is checks_second.py, both under _Local/Erdos_problems/UnsolvedMath/attempt_audit_2026_10_01/number_theory/. The result is a complete classification only within the stated bound. No interval beyond $5000$ has been excluded by this calculation.

Remaining gap and outcome.

The attempt supplies exact inverse-class witnesses and a reproducible finite test respecting the order discriminant. It supplies no effective global bound and no uniform construction of a nonambiguous class at all larger discriminants. Outcome: UNRESOLVED. The finite count of $65$ is not promoted to an unconditional proof of completeness.

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1252 (NT-045), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Peter J. Weinberger, Exponents of the class groups of complex quadratic fields, Acta Arithmetica 22 $1973$, 117-124. https://eudml.org/doc/205250. Bibliographic information from the supplied record.

• S3 Ernst Kani, Idoneal Numbers and Some Generalizations, Annales des sciences mathematiques du Quebec 35 $2011$, 197-227. https://mast.queensu.ca/~kani/papers/2011K1.pdf. Bibliographic information from the supplied record.

• E1 Ernst Kani, Idoneal Numbers and some Generalizations, author manuscript. https://mast.queensu.ca/~kani/papers/idoneal-f.pdf. Author-manuscript formulation and search-visible text inspected 1 October 2026; direct PDF retrieval was unavailable. The standard reduction and inverse-class facts are explicitly identified as inputs.