Giuga Number¶

A composite integer n for which every prime divisor p satisfies p dividing n/p minus 1, a restrictive factorwise congruence linked to Giuga's primality conjecture.

Version: v3 · 2026-09-07 · History
Domain-specific #: 1937
Origin domain: number theory
Subdomain: special classes of composite integers
Aliases: Giuga composite

Core Idea¶

A Giuga number is a positive composite integer \(n\) such that, for every prime divisor \(p\) of \(n\),

\[ p\mid\left(\frac np-1\right). \]

Equivalently, every prime divisor satisfies \(p^2\mid n-p\). This factor-by-factor condition defines a sparse and structurally constrained class of composite numbers.

The smallest example is \(30=2\cdot3\cdot5\). For \(p=2,3,5\), the corresponding values \(30/p-1\) are \(14,9,5\), each divisible by \(p\). The test is universal over all prime divisors: satisfying it for all but one factor is failure.

The class matters beyond cataloging unusual integers. Giuga's primality conjecture can be restated using power sums, and the 1996 analysis by Borwein, Borwein, Borwein, and Girgensohn shows that a composite counterexample to the relevant primality congruence would have to be both a Carmichael number and a Giuga number^[1]. Thus Giuga numbers isolate one of two independent factorwise conditions behind a possible pseudoprime.

Structural Signature¶

The mandatory roles are:

a positive integer \(n\);
proof that \(n\) is composite;
the set of distinct prime divisors \(p\mid n\);
the quotient \(n/p\) for each such divisor;
the congruence \(n/p\equiv1\pmod p\) for every divisor; and
equivalent global or arithmetic formulations only when their hypotheses are satisfied.

The recognition test is finite once the prime factorization is known. For each distinct prime factor, compute the cofactor and reduce it modulo that prime. A single nonzero residue rejects the candidate.

The definition itself implies squarefreeness. If \(p^2\mid n\), then \(n/p\equiv0\pmod p\), so \(n/p-1\equiv-1\pmod p\), contradicting the defining condition. Squarefreeness is therefore a consequence, not an optional extra criterion.

What It Is Not¶

A Giuga number is not a prime number. Compositeness is explicitly part of the definition even though the class arises in a primality-conjecture context.

It is not a Carmichael number. Korselt's criterion for a squarefree Carmichael number requires \(p-1\mid n/p-1\) for every prime divisor \(p\); Giuga requires \(p\mid n/p-1\)^[1]. The divisors \(p-1\) and \(p\) create different classes.

It is not automatically a counterexample to Giuga's conjecture. The Borwein analysis requires the intersection of the Giuga and Carmichael conditions for the relevant composite counterexample. An ordinary Giuga number such as 30 does not by itself refute the conjecture.

It is not the same as a primary pseudoperfect number, whose reciprocal-prime relation has a fixed equality. Giuga's equivalent reciprocal condition requires a positive integer value; known small examples often produce 1, but the definitions and open questions must not be collapsed.

It is not any squarefree composite number. Squarefreeness is necessary, while the cofactor congruence for every prime divisor is much stronger.

Scope of Application¶

The node applies in elementary and computational number theory, studies of primality criteria, special integer sequences, congruences, reciprocal-prime identities, and arithmetic derivatives.

Its direct use is classification: after factoring a composite integer, researchers can decide membership with the factorwise congruence. Equivalent forms can support proofs and searches. The reciprocal characterization relates the local conditions to a global rational expression, while power-sum and Bernoulli-number formulations connect the class to proposed primality characterizations^[2].

The class also supplies a boundary object in pseudoprime research. Carmichael numbers satisfy one factorwise divisibility pattern; Giuga numbers satisfy another. Their possible intersection is far more constrained than either class alone.

Statements about how many examples are currently known, the largest known term, completeness to a bound, parity of all known examples, or lower bounds on hypothetical odd examples are discovery-state claims. They should be cited and dated rather than built into the invariant.

Clarity¶

For \(n=30\):

\(30/2-1=14\), divisible by 2;
\(30/3-1=9\), divisible by 3; and
\(30/5-1=5\), divisible by 5.

Thus 30 passes.

For \(n=60\), the prime divisor 2 fails: \(60/2-1=29\), which is not divisible by 2. The repeated factor also triggers the general squarefreeness contradiction.

For \(n=42=2\cdot3\cdot7\), the divisor 3 fails because \(42/3-1=13\not\equiv0\pmod3\). A candidate need not have repeated factors to fail.

The expression “for every prime divisor” means distinct primes in the logical quantifier. Multiplicity enters indirectly because any repeated prime factor makes the relevant condition impossible.

Manages Complexity¶

The definition compresses many congruence checks into a named integer class. Instead of repeatedly saying “composite, squarefree, and every prime cofactor is 1 modulo that prime,” a proof can invoke Giuga membership and use established equivalences.

It also decomposes a difficult global primality-congruence question into local arithmetic constraints. Factorwise reasoning exposes why squarefreeness follows, why semiprimes cannot work, and why intersection with Carmichael numbers is restrictive.

Equivalent formulations provide different computational and theoretical handles. Congruences support modular testing; reciprocal-prime expressions expose additive structure among factors; arithmetic-derivative equations connect the same membership relation to another arithmetic operation.

This compression must not hide dependence on factorization. The elementary membership test is easy given the complete prime factorization, but obtaining that factorization for a large integer is a separate computational problem.

Abstract Reasoning¶

Squarefree inference. If \(p^2\mid n\), the cofactor \(n/p\) is divisible by \(p\), contradicting \(n/p\equiv1\pmod p\).

No semiprimes. Suppose \(n=pq\) with primes \(p<q\). Applying the condition at \(q\) requires \(q\mid p-1\), impossible because \(0<p-1<q\). Hence a Giuga number has at least three distinct prime factors.

Local-to-global reciprocal form. For squarefree \(n=\prod p\), summing cofactors \(n/p\) and comparing modulo each prime leads to the integer condition

\[ \sum_{p\mid n}\frac1p-\frac1n\in\mathbb N. \]

This is a theorem-level equivalence, not a replacement to quote without compositeness and factor hypotheses.

Intersection reasoning. A composite satisfying the Giuga power-sum primality congruence must meet both Giuga's \(p\)-divisibility and Carmichael's \(p-1\)-divisibility constraints. Membership in only one class is insufficient.

Knowledge Transfer¶

The portable skeleton is classification by a universal local constraint: factor an object into irreducible constituents, test a cofactor relation at every constituent, and name the objects that satisfy all tests.

That skeleton appears elsewhere in algebra and computation, but literal Giuga identity is inseparable from composite integers, prime divisors, modular arithmetic, and the exact cofactor congruence. A network whose every component passes a local check is not a Giuga number.

The class also illustrates how a named abstraction can turn an unwieldy conjunction into a reusable premise. The transfer belongs to Classification; the mathematical residual remains domain-specific.

Examples¶

30. The factors 2, 3, and 5 all pass, making 30 the smallest Giuga number^[3].

858. Its factorization \(2\cdot3\cdot11\cdot13\) passes the factorwise test. For example, \(858/13-1=65\), divisible by 13.

1722. With factors \(2\cdot3\cdot7\cdot41\), each cofactor minus one is divisible by its associated factor; for 41, \(1722/41-1=41\).

60 as a nonexample. Its repeated factor 2 is enough to force failure, and direct calculation gives 29 modulo 2.

561 as a boundary example. It is a Carmichael number, but it is not thereby a Giuga number^[4]. The two factorwise predicates must be checked separately.

Structural Tensions¶

Local definition versus global consequences. Membership is checked prime by prime, yet it yields reciprocal identities and constraints on primality congruences.

Simple test versus hard candidate generation. The modular condition is elementary after factorization, while finding large members and proving search bounds can be difficult.

Named class versus open census. The definition is exact and timeless, but counts of known members and claims about parity or infinitude can change.

Equivalent formulas versus hidden hypotheses. Congruence, reciprocal, Bernoulli, and derivative forms expand the toolkit, but casual restatement can drop compositeness, squarefreeness, or quantifier scope.

Relation to primality versus composite identity. The class is important because of a primality conjecture, yet every member is composite by definition.

Structural–Framed Character¶

Giuga Number is structurally crisp but strongly mathematically framed. It has a short recognition test that applies across all candidate integers, and it supports proofs independent of notation or computing platform. However, its indispensable objects are integers, prime factorizations, divisibility, congruence, and arithmetic functions.

It is domain-specific rather than prime. The transferable pattern of universal classification is already generic; the exact \(p\mid n/p-1\) condition has no substrate-neutral identity.

Structural Core vs. Domain Accent¶

The structural core is: decompose a candidate into irreducible factors, impose one local relation for every factor, and classify the candidate only if the universal conjunction holds.

The domain accent is the entire arithmetic content: positive composite integer, prime divisor, quotient cofactor, modular congruence, squarefreeness consequence, reciprocal characterization, and relation to Carmichael numbers and Giuga's conjecture.

`prime:classification` owns the reusable act of assigning cases by defining predicates. It does not make the Giuga predicate redundant, because the latter supplies a stable named class with nontrivial equivalences and consequences.

Giuga Number specializes Classification: membership is an exact predicate over composite integers, and the name packages that predicate as a mathematical class. This is the minimal proposed DAG parent.

It relies on prime factorization, divisibility, modular arithmetic, and universal quantification. Those are mathematical tools or logical relations, not additional taxonomic parents.

It is related to verification because a certificate can list the prime factors and one congruence check per factor. It is related to conjunction because every local condition must hold. These explanatory relations remain in prose.

Relationships to Other Abstractions¶

Current abstraction Giuga Number Domain-specific

Parents (1) — more general patterns this builds on

Giuga Number is a kind of Classification Prime

Giuga Number specializes Classification: membership is an exact predicate over composite integers, and the name packages that predicate as a mathematical class.

Hierarchy path (1) — routes to 1 parentless root

Giuga Number → Classification

Neighborhood in Abstraction Space¶

Giuga Number sits in a sparse region of the domain-specific corpus (70^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Computational Number Theory & Enumeration (13 abstractions)

Nearest neighbors

Primefree Sequence — 0.88
Divisor Function — 0.86
Quartic reciprocity — 0.84
Descartes Number — 0.83
Fundamental Theorem of Arithmetic — 0.83

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Carmichael number. Requires \(p-1\mid n/p-1\), not \(p\mid n/p-1\).

Primary pseudoperfect number. Defined by a specific reciprocal-prime equality; related but not generally identical.

Strong Giuga number. Terminology used for a more restrictive intersection or conjecture-related condition; it should not replace ordinary Giuga membership.

Giuga's conjecture / Agoh–Giuga conjecture. A primality characterization, not the integer class itself.

Squarefree composite. A necessary superclass property without the defining cofactor congruences.

Giuga sequence. A construction used in searching or generating related numbers, not automatically the same as a Giuga number.

References¶

[1] Borwein, et al. “Giuga's Conjecture on Primality”. The American Mathematical Monthly, 1996. For both halves: the power-sum restatement of Giuga's conjecture and the paper's Theorem 1, which makes any composite counterexample simultaneously a Carmichael number and a Giuga number. For the side-by-side contrast: the paper states Korselt's 1899 criterion in the same cofactor form p-1 | n/p-1 that the article uses, alongside the Giuga condition p | n/p-1. registry ↩a ↩b

[2] Agoh. “On Giuga’s conjecture”. Manuscripta Mathematica, 1995. For the Bernoulli-number formulation and its connection to the proposed primality characterization; the reciprocal characterization in the first half of the sentence is Giuga's identity as given by Borwein et al., not Agoh's. registry ↩

[3] OEIS Foundation Inc. A007850: Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n. The On-Line Encyclopedia of Integer Sequences (OEIS), maintained by The OEIS Foundation Inc., 2026. For 30 as the least term of the sequence of Giuga numbers, with the entry's own verification that 2, 3 and 5 each satisfy p | n/p - 1; accessed 2026, the entry being a living record. registry ↩

[4] Crandall, Richard and Pomerance, Carl. Prime Numbers: A Computational Perspective, 2^nd edition. Springer, 2005. For 561 as a Carmichael number (Section 3.4.2); that it is not thereby a Giuga number follows from the definition given earlier in the article and is not in this book. registry ↩