Argam is a naming system for integers based on prime factorization. Every positive integer has a unique Argam name derived from its prime factors, their exponents, and a set of canonical combining rules.

Numbers 1–720 have direct lookup names (720-smooth). Numbers above 720 are built compositionally from prime prefixes, power suffixes, and junction rules.

The maximum supported value is 2¹⁰⁰ = 1,267,650,600,228,229,401,496,703,205,376. Primes 2–13,999 have individually named forms (Chapters 1–2). Primes 14,009–1,299,709 are named via the Chapter 3–5 positional system. Primes 1,299,721–2,750,159 are named via the Chapter 6 positional system. Primes 2,750,161–7,368,787 are named via the Chapter 7 positional system. Primes 7,368,791–15,485,863 are named via the Chapter 8 positional system. Primes above 15,485,863 have no named form and are shown as [p].

Negative integers from -1 to -10 have dedicated names, formed by combining the number's prefix with the suffix -zalt:

The minimum supported value is -10 (Dezzalt). Values below -10 are not supported.

Each prime up to 13,597 has a dedicated short prefix used when building composite names. Examples:

Primes 14,009–1,299,709 named via Chapters 3–5. Primes 1,299,721–2,750,159 via Chapter 6. Primes 2,750,161–7,368,787 via Chapter 7. Primes 7,368,791–15,485,863 via Chapter 8. Primes above 15,485,863 are shown as [p].

When the greatest prime factor of a number > 720 is 13, 17, 23, 29, or 31, that prime is replaced by a suffix appended to the base name (the remaining factors must be ≤ 720):

If a larger prime is present, the suffix is not used. Instead, the prime takes a modified prefix as a standalone token:

The suffix -ode is no longer used for 31; it is replaced entirely by -sode. Vowel junction rules apply between the base name and the suffix.

When a prime is raised to a power, a suffix replaces or appends to its prefix:

If the base name ends in a vowel and the suffix starts with a vowel, the trailing vowel of the base is dropped before appending.

When a number equals 2 × (perfect power) or 3 × (perfect power), an alternative name ending in -di or -tri may be constructed.

Excluded bases: powers of 2, powers of 3, or powers of 6 (they already have dedicated names). The -di/-tri form is shown as an Alt name alongside the primary composite name.

1. Factorize n into prime powers: p₁^e₁ × p₂^e₂ × …
2. Check if the entire number is itself a perfect power (exponents share a GCD > 1). If so, name the base and apply the power suffix.
3. Find the greatest prime factor. If it is 13, 17, 23, 29, or 31 and the remaining base ≤ 720, apply the prime suffix (G.5–G.6).
4. Otherwise, greedily group small prime factors (≤ 31) into the largest nameable composites (≤ 720) that have canonical names. Remaining large primes become individual tokens, using modified prefixes (flor-, nev-, sode-, zote-) when a larger prime is present.
5. Sort tokens by their numeric value (ascending).
6. Concatenate tokens left-to-right, dropping a trailing vowel from the left part when the right part starts with a vowel.
7. Capitalize the result.

These primes are structural milestones in the Argam system — every 100th named prime and the special prime 647. They are highlighted in the Prime Explorer.

The maximum supported converter value is 2¹⁰⁰. The last hand-named prime is 13,999 (herge-). Chapters 3–8 cover 14,009–15,485,863. Primes above 15,485,863 are shown as [p].