Highly totient number

An integer whose number of preimages under Euler’s totient function exceeds that of every smaller integer.

Version

v1 · 2026-09-08 · History

Domain-specific #

4883

Origin domain

number theory

Subdomain

number theory

Core Idea

The record is over output values rather than input size, nontotients have zero preimages and ties do not create a new highly totient record. For each candidate k, all integers x satisfying phi of x equals k are counted, and k qualifies when this multiplicity sets a strict running maximum over positive integers below it. 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.

Scope of Application

Highly totient number belongs to number theory and is useful where the analyst can specify the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the positive integer k, Euler totient function, solution set of phi of x equals k, finite preimage count, strict comparison with every smaller integer, initial values and multiplicities, parity and infinitude properties and relation to highly composite numbers and nontotients are explicit.

Clarity

The abstraction clarifies a crowded vocabulary by making the positive integer k, Euler totient function, solution set of phi of x equals k, finite preimage count, strict comparison with every smaller integer, initial values and multiplicities, parity and infinitude properties and relation to highly composite numbers and nontotients 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.

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 Highly totient number. Highly totient 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

1. Identify the carrier. State what the elements, states, objects, or observations are: the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the positive integer k, Euler totient function, solution set of phi of x equals k, finite preimage count, strict comparison with every smaller integer, initial values and multiplicities, parity and infinitude properties and relation to highly composite numbers and nontotients are explicit independently of one notation or implementation.

Knowledge Transfer

Knowledge transfers strongly among subfields of number theory because they reuse the typed number theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, For each candidate k, all integers x satisfying phi of x equals k are counted, and k qualifies when this multiplicity sets a strict running maximum over positive integers below it., and type the carrier, state every parameter and convention in the definition, test that the positive integer k, Euler totient function, solution set of phi of x equals k, finite preimage count, strict comparison with every smaller integer, initial values and multiplicities, parity and infinitude properties and relation to highly composite numbers and nontotients are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions

Current abstraction Highly totient number Domain-specific

Parents (1) — more general patterns this builds on

• Highly totient number is a kind of Optimization Prime

  The proposed strict upward parent is prime:optimization.

Hierarchy path (1) — routes to 1 parentless root

• Highly totient number → Optimization

Neighborhood in Abstraction Space

Highly totient number sits in a crowded region of the domain-specific corpus (2^nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Number-Theoretic Sequences & Classes (37 abstractions)

Nearest neighbors

• Nonhypotenuse number — 0.94
• Arithmetic function — 0.94
• Unusual number — 0.94
• Euler's totient function — 0.94
• Highly composite number — 0.94

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