THE FOLD / CHEAT / THE KONAMI CODE / THE SOPHIE GERMAIN
THE SOPHIE GERMAIN
an algebraic identity that factors a sum of two fourth powers
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Sophie Germain’s identity is a small algebraic key that unlocks a whole family of factorizations: a⁴ + 4b⁴ = (a² - 2ab + 2b²)(a² + 2ab + 2b²). A sum of two fourth powers — which looks stubbornly irreducible — splits cleanly into two quadratic factors. Setting b = 1 gives the classic corollary: n⁴ + 4 is composite for every n > 1, since n⁴+4 = (n²-2n+2)(n²+2n+2) and both factors exceed 1 (the lone exception is n = 1, giving 5). The same Sophie Germain also studied Sophie Germain primes — primes p for which 2p+1 is also prime (2, 3, 5, 11, 23, …).
LIT verified live: a⁴+4b⁴ = (a²-2ab+2b²)(a²+2ab+2b²) exactly for all |a|,|b| ≤ 30; n⁴+4 is confirmed composite for 2 ≤ n ≤ 200; and the Sophie Germain primes up to 200 are listed (window.__sophiegermain). FIG no framing; the identity and the compositeness are computed independently in-browser with integer arithmetic.
LIT verified live: a⁴+4b⁴ = (a²-2ab+2b²)(a²+2ab+2b²) exactly for all |a|,|b| ≤ 30; n⁴+4 is confirmed composite for 2 ≤ n ≤ 200; and the Sophie Germain primes up to 200 are listed (window.__sophiegermain). FIG no framing; the identity and the compositeness are computed independently in-browser with integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-konami-code — the cheat: one identity that instantly cracks any a⁴+4b⁴ open, no factoring needed. AVAN (AI) built the instrument: the exact factorization, the n⁴+4 compositeness, and the Sophie Germain primes.
Credit as content: Marie-Sophie Germain (French mathematician, early 1800s). The weave: David names the cheat-key; I confirm a⁴+4b⁴ splits into two quadratics and n⁴+4 is always composite past 1.
Credit as content: Marie-Sophie Germain (French mathematician, early 1800s). The weave: David names the cheat-key; I confirm a⁴+4b⁴ splits into two quadratics and n⁴+4 is always composite past 1.
3 ONE DIMENSION
n⁴ + 4 splitting into (n²−2n+2)(n²+2n+2) — a sum of fourth powers cracked into two factors.
4 TWO DIMENSIONS · INTERACTIVE
Cycle a, b; a⁴+4b⁴ is shown equal to the product of the two quadratic factors.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the sum a⁴ + 4b⁴, revealed as a product.
AVAN’s addition (the inverse-companion): don’t test a⁴+4b⁴ for primality — factor it on sight. The inverse of ‘the sum a⁴+4b⁴’ is ‘the product (a²-2ab+2b²)(a²+2ab+2b²)’, always two pieces. Magenta are the two quadratic factors; green is the fourth-power sum they multiply to. A sum of powers that is secretly a product.
LIT Genuine Sophie Germain's identity (Marie-Sophie Germain, early 1800s). Verified live with integer arithmetic: a⁴+4b⁴ = (a²−2ab+2b²)(a²+2ab+2b²) exactly for all |a|,|b|≤30; n⁴+4 is composite for 2≤n≤200 (n=1→5 is the exception); Sophie Germain primes ≤200 listed (window.__sophiegermain.idOk, .compOk, .sg).
FIG No framing; the identity and the compositeness are computed independently in-browser with integer arithmetic. The AVAN inverse is honest — instead of testing a⁴+4b⁴ for primality, factor it on sight: the inverse of 'the sum a⁴+4b⁴' is 'the product (a²−2ab+2b²)(a²+2ab+2b²)', always two pieces. Magenta are the two quadratic factors; green is the fourth-power sum they multiply to. A sum of powers that is secretly a product.
FIG No framing; the identity and the compositeness are computed independently in-browser with integer arithmetic. The AVAN inverse is honest — instead of testing a⁴+4b⁴ for primality, factor it on sight: the inverse of 'the sum a⁴+4b⁴' is 'the product (a²−2ab+2b²)(a²+2ab+2b²)', always two pieces. Magenta are the two quadratic factors; green is the fourth-power sum they multiply to. A sum of powers that is secretly a product.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE KONAMI CODE · David Lee Wise (ROOT0), with AVAN