Amenable number¶

A positive integer n admitting a multiset of exactly n integers whose sum and product are both n.

Version: v2 · 2026-09-19 · History
Domain-specific #: 3270
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomain: Recreational Number Theory → Mathematics

Core Idea¶

Negative and repeated integers are allowed, and the characterization selects numbers congruent to zero or one modulo four except four under the stated convention. Pairs of one and negative one preserve product while adjusting length and sum, and arithmetic constructions build a witness multiset for each admissible congruence class. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

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The Add-and-Multiply Puzzle

Pick a number, like 5. Now try to find exactly 5 numbers that add up to 5 and also multiply to 5. You are allowed to use the same number again and below-zero numbers too: 5, 1, 1, minus 1, minus 1 works. A number where a puzzle like this can be solved is called an amenable number.

Add-Equals-Multiply Numbers

A number n is amenable if you can pick exactly n whole numbers, where repeats and negative numbers are allowed, so that they add up to n and also multiply to n. For example, 5 works: take 5, 1, 1, -1 and -1. Adding gives 5, and multiplying also gives 5, because the two minus ones cancel out. The trick is that adding extra 1s, or pairs of -1s, changes how many numbers you have and what they add up to, without changing what they multiply to. It turns out the numbers that work are exactly those that leave a remainder of 0 or 1 when divided by 4, except the number 4 itself.

Sum-Equals-Product Integers

An amenable number is a positive integer n for which there exists a multiset of exactly n integers, with repetition and negative values allowed, whose sum and product both equal n. For example, 5 is amenable because 5 + 1 + 1 + (-1) + (-1) = 5 and 5 × 1 × 1 × (-1) × (-1) = 5. The useful building blocks are 1s and pairs of -1s: they change the length and the sum of the list while keeping the product the same. Using such constructions, mathematicians characterized the amenable numbers: they are exactly the numbers that leave remainder 0 or 1 when divided by 4, except for 4 itself, under the stated conventions.

An amenable number is a positive integer n such that some multiset of exactly n integers, with repetition and negative entries permitted, has both sum and product equal to n. The characterization, under the stated conventions, is that n is amenable precisely when n ≡ 0 or 1 (mod 4), with the exception n = 4. Sufficiency is shown by constructing a witness multiset for each admissible congruence class: padding with 1s and with pairs of -1s preserves the product while adjusting length and sum, so arithmetic choices of a few other factors can hit the required totals. For instance, n = 5 has witness {5, 1, 1, -1, -1}. The definition is sensitive to its parameters: cardinality must be exactly n, entries are integers, both sum and product must match, and the sign convention and exceptional case must be stated. Changing any of these, for example forbidding negatives, defines a different problem.

Scope of Application¶

Amenable number belongs to recreational number theory and is useful where the analyst can specify the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit. The scope is broad within that domain but bounded by the need for the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Amenable number can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Amenable number. Amenable number compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of recreational number theory because they reuse the typed recreational number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Pairs of one and negative one preserve product while adjusting length and sum, and arithmetic constructions build a witness multiset for each admissible congruence class., and type the carrier, state every parameter and convention in the definition, test that the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Amenable number Domain-specific

Parents (1) — outgoing typed relations

Amenable number is a kind of Constraint Prime

It remains its own entry because its identity is fixed by the positive integer n, multiset cardinality exactly n, integer entries and repetition, sum equality, product equality, sign convention, congruence characterization, exceptional cases and witness construction.

Hierarchy path (1) — routes to 1 parentless root

Amenable number → is a kind of Constraint

Neighborhood in Abstraction Space¶

Amenable number sits in a crowded region of the domain-specific corpus (9^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Numeral Systems & Integer Sequences (22 abstractions)

Nearest neighbors

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