Luhn formula¶

The Luhn formula is the decimal modulus-10 check-digit algorithm that alternately doubles digits, reduces two-digit products, and selects a final digit making the transformed sum divisible by ten.

Version: v1 · 2026-09-28 · History
Domain-specific #: 7690
Origin domain: Mathematics Logic Statistics

Core Idea¶

The Luhn formula, also called the Luhn algorithm, modulus 10, or mod 10 algorithm, is a decimal check-digit procedure for detecting accidental errors in identification numbers. It appends one digit chosen so that a weighted transformation of the complete number has a sum divisible by ten. To calculate the check digit, scan the payload from right to left and double every second digit, beginning with the digit in the appropriate alternating position for an appended check digit.

Scope of Application¶

The Luhn formula applies to decimal identifiers whose issuer has adopted its terminal check-digit convention: alternating right-to-left doubling, subtract-nine reduction, and a transformed sum divisible by ten. Its reach is limited to accidental-error screening and internal consistency; each identifier scheme must separately establish that it uses ordinary Luhn rather than another mod-10 or modified rule.

Payment-card numbers — generate and validate the decimal check digit under the issuer-numbering convention.
IMEI equipment identifiers — screen entered device numbers for consistency with the adopted Luhn position and parity.
CUSIP identifiers — apply the relevant Luhn-based check convention while respecting the identifier's declared character handling.
United States National Provider Identifiers — validate the decimal check digit under the scheme's prescribed Luhn treatment.

Clarity¶

Passing a Luhn check means that a decimal identifier is internally consistent with one check-digit rule; it does not mean that the issuer assigned it, the account exists, or the presenter is authorized to use it. That distinction keeps transcription screening separate from authentication and cryptographic integrity. A deliberately fabricated number can easily be given a valid Luhn digit.

Manages Complexity¶

Identification schemes vary in issuer, length, grouping, and business meaning, but their decimal transcription check can be reduced to a payload, a check-digit position, an alternating parity, a digit transform, and one modulus-10 remainder. Implementers need not encode separate error logic for every identifier family: scanning from the check-digit end, doubling the designated positions, reducing products above nine, and summing yields either the required check digit or a single validity result.

Abstract Reasoning¶

Luhn reasoning maps a decimal digit string to one modular invariant. From a payload with a declared check-digit position, to the check digit, the algorithm fixes right-to-left parity, doubles alternating digits, subtracts nine from products above nine, and selects the digit that makes the transformed sum congruent to zero modulo ten. Running the same transform in reverse use—from a complete identifier to a validity result—tests consistency with that rule without consulting the identifier's issuer or record system.

Knowledge Transfer¶

Within identifier validation, the Luhn formula transfers literally across account, card, and reference-number systems that adopt its decimal check-digit convention. The cargo that carries intact is the payload digits, check-digit position, right-to-left alternating parity, doubling transform, subtraction of nine for products above nine, and divisibility-by-ten invariant. Diagnostics transfer by recomputing the check digit, testing a complete identifier, and locating parity or digit-order mistakes. This is (C) a formal algorithm wherever that exact convention is used.

Relationships to Other Abstractions¶

Current abstraction Luhn formula Domain-specific

Parents (1) — more general patterns this builds on

Luhn formula is a kind of Luhn Mod N Algorithm Domain-specific

Luhn Formula strictly instantiates the general check-character scheme with the ordered alphabet fixed to decimal digits, N fixed to 10, the selected-position operation fixed to doubling followed by subtract-nine reduction, and a terminal digit chosen to complete the zero congruence.

Hierarchy paths (2) — routes to 2 parentless roots

Luhn formula → Luhn Mod N Algorithm → Algorithm → Function (Mapping)

Show alternative path (1)

Neighborhood in Abstraction Space¶

Luhn formula sits in a sparse region of the domain-specific corpus (86^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2551 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08