Departure question
Find order without counting everythingPort 7 of 16Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
This voyage is an editorial path for understanding, not a claim of direct historical influence or sole invention.
Integers belong to the same class when they leave the same remainder modulo n. This appears naturally in clocks, calendars, checksums, and number theory; Gauss systematized congruence notation and theory in 1801. RSA and Diffie–Hellman use modular arithmetic centrally, while hashes, error correction, and internet security also require other algebra and protocols.
× | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
2 | 0 | 2 | 4 | 6 | 1 | 3 | 5 |
3 | 0 | 3 | 6 | 2 | 5 | 1 | 4 |
4 | 0 | 4 | 1 | 5 | 2 | 6 | 3 |
5 | 0 | 5 | 3 | 1 | 6 | 4 | 2 |
6 | 0 | 6 | 5 | 4 | 3 | 2 | 1 |
Multiplication table mod 7 — every nonzero row contains exactly one 1, a consequence of 7 being prime.
The arithmetic of clocks — wrapping around at a modulus. The basic language of number theory and cryptography.
Q.7 o’clock + 8 hours = ?
Q.3 × 5 (mod 7)
Ports in time
Follow the scenes to see problems, notation, standards of proof, and applications changing across different times and places.
Euclid already used the greatest-common-divisor algorithm. The tools of modular arithmetic existed long before its formal notation.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearAt age 24, Gauss introduced congruence notation (≡) and organized modular arithmetic into a general theory, transforming number theory.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearThe congruence a^p ≡ a (mod p) for prime p became a core tool of public-key cryptography more than three centuries later.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearRon Rivest, Adi Shamir, and Leonard Adleman built public-key encryption from modular exponentiation and the difficulty of factoring a product of large primes.
No reliable place is given, so time continues without an invented pin
Continue through the world of this yearRSA, Diffie–Hellman, and elliptic-curve cryptography; hash functions; checksums; and twelve-tone music theory.
Curated sources and problems. Bring one discovery back from OEIS, Project Euler, MathOverflow, or arXiv.
No concept belongs to one person
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Number lenses
These numbers are editorial lenses for the voyage, not required prerequisites.
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Trace recurring structure through exploding possibilities, divisibility, and the apparent irregularity of primes.
Open the number voyage
Concept genealogy
Concepts arriving from before
Current port
Modular Arithmetic
Only direct editorial links are shown; this is not a complete learning order or historical influence line.
