Take the Gaussian integer 2^n + 3i in Z[i]. Its squared norm is |2^n + 3i|^2 = 4^n + 9. The first value: |2 + 3i|^2 = 4 + 9 = 13, which is prime. This family of norms connects powers of 2, the prime 3, and Gaussian arithmetic. Every prime factor of any value must be congruent to 1 mod 4 (Fermat's theorem on sums of coprime squares).
|2^n + 3i|^2 is prime at exactly these positions:
Prime positions: {1, 3, 5, 9, 15}. Gaps between positions: {2, 2, 4, 6}. All prime positions are odd -- even n always gives a multiple of 5.
5 systematically blocks even indices. Only odd n can produce primes.
Each covering prime q divides 4^n + 9 for some n. Restricted to odd n, the first 6 covering primes and their periods:
| Prime q | Period | First odd n |
|---|---|---|
| 13 | 6 | 1 |
| 37 | 18 | 17 |
| 61 | 30 | 21 |
| 73 | 9 | 3 |
| 97 | 24 | 7 |
| 181 | 90 | 11 |
All six periods factor into {2, 3, 5} only. The first odd residues {1, 3, 7, 11, 17, 21} all factor into chain primes (21 = 3*7).
Three-way classification: ODD-covering 25%, EVEN-covering 28%, NON-covering 47%. Densities stable across all bounds tested.
Enter n to compute |2^n + 3i|^2 = 4^n + 9. Check primality and the mod-5 filter.
Enter n (1-25):
Try: n=1 (13, prime), n=3 (73, prime), n=5 (1033, prime), n=9 (prime), n=2 (even, blocked by 5).
| Aspect | Standard view | Ring structure |
|---|---|---|
| Gaussian norms | Z[i] norms factor into primes 1 mod 4 | The family 4^n + 9 has prime values at n = 1, 3, 5, 9, 15 -- all odd |
| First value | 4 + 9 = 13, the sixth prime | |2 + 3i|^2 = 13, a chain prime of Z/214,414,200 |
| Covering sum | 462 is a sum of six primes | 462 = 2 * 3 * 7 * 11. Factors into chain primes; 5 is absent |
| Mod-24 | Quadratic residue classification | 13 mod 24 selects odd-covering primes, 5 mod 24 selects even-covering |
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