The genetic code maps 64 codons to 21 targets (20 amino acids + stop). Since 21 = 3 * 7 and gcd(3, 7) = 1, the Chinese Remainder Theorem gives Z/21 = Z/3 x Z/7. The mod-3 channel correlates with chemical class. The mod-7 channel correlates with size. Codon degeneracy takes exactly the values {1, 2, 3, 4, 6} -- each divides phi(21) = 12.
Codon Table Structure
Structure
Count
Factorization
Nucleotide bases
4
2^2 (A, T/U, G, C)
Codons
64
2^6 (triplets of 4 bases)
Amino acids
20
2^2 * 5
Stop codons
3
3 (UAA, UAG, UGA)
Sense codons
61
64 - 3 (prime)
Amino acid targets
21
3 * 7 (20 amino acids + 1 stop group)
The key number is 21: the number of distinct targets the genetic code must distinguish. 21 = 3 * 7, and since gcd(3, 7) = 1, the CRT decomposition Z/21 = Z/3 x Z/7 applies. This gives two independent channels.
CRT Decomposition: Z/3 x Z/7
Codon Ring Decomposition
Z/21 = Z/3 x Z/7 by CRT (gcd(3,7) = 1). If you number the 21 targets 0-20, each has unique coordinates (residue mod 3, residue mod 7). An observed correlation: mod-3 classes align with amino acid chemical polarity, while mod-7 classes align with molecular size. This is suggestive but not proved to be structural -- the alignment depends on the numbering scheme.
Codon degeneracy -- how many codons encode each amino acid -- takes the values {1, 2, 3, 4, 6}. These are the proper divisors of phi(21) = phi(3) * phi(7) = 2 * 6 = 12. The value 5 cannot appear because 5 does not divide 12. The value 12 does not appear either -- no amino acid has 12-fold redundancy.
Degeneracy 1
Met (M), Trp (W)
Unique codons. No redundancy. The rarest amino acids.
Degeneracy 2
9 amino acids
2-fold coverage. Wobble at position 3.
Degeneracy 3
Ile (I)
3-fold coverage. Three codons for isoleucine.
Degeneracy 4
5 amino acids
4-fold. Full wobble class.
Degeneracy 6
Leu (L), Arg (R), Ser (S)
6-fold. Maximum redundancy.
Wobble: Error Tolerance
Point mutations at codon position 3 are silent about 2/3 of the time. The third base 'wobbles' -- multiple bases encode the same amino acid. This is biological error tolerance built into the code's redundancy:
Position 1
~0% silent
First base mutations almost always change the amino acid.
Position 2
~5% silent
Second base: chemical class usually changes.
Position 3
~67% silent
Wobble position. Most mutations are invisible to the protein.
Explore: Codon Ring Z/21
Enter a codon number (1-64). See its position in Z/3 x Z/7. The mod-3 channel correlates with chemical class. The mod-7 channel correlates with size.