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phi^2 + 1/phi^2 = 3, checked every way I could think of

2026-08-11 · 5 min read

An exact identity, six proof steps machine-verified, and a search over 1,476,000 candidates that found no other root — plus what the identity does not license.

The identity is two lines from the definition. What follows is what happened when I stopped trusting that and checked it mechanically instead.

The identity

phi = (1 + sqrt(5)) / 2
phi^2 = phi + 1        (the defining quadratic)
1/phi = phi - 1
1/phi^2 = (phi-1)^2 = 2 - phi

phi^2 + 1/phi^2 = (phi + 1) + (2 - phi) = 3

phi cancels identically. This is exact algebra in Q(sqrt 5), not a numerical near-hit.

Checked four ways

• Decimal at 80 digits: the sum differs from 3 by 1e-79, which is the rounding of an irrational division and nothing else.
• mpmath at 60, 210 and 1000 digits: the difference is exactly 0.0 at every precision.
• sympy symbolically: simplify(phi2 + 1/phi2) returns the integer 3. There is no residual to hide at any precision.
• Every one of the six written steps verified independently — so the proof is not a correct conclusion reached by a broken argument.

And the converse, which is the part worth having

Solving x^2 + 1/x^2 = 3 gives exactly four roots: phi, 1/phi, -phi, -1/phi. Clearing denominators gives x^4 - 3x^2 + 1, which factors as (x^2-x-1)(x^2+x-1).

Since the expression is invariant under x -> -x and under x -> 1/x, phi is the unique solution up to the expression’s own symmetry group. The identity pins down phi rather than merely holding for it.

A numerical net was run as well — 1,476,000 candidates of the form (p + q*sqrt(d))/r across p, q, d, r ranges — and produced no non-golden solution, as the degree-4 factorisation requires.

What it does not license

The identity is logically equivalent to the definition, not additional to it: x^4-3x^2+1 factors into the two quadratics, so "x^2+1/x^2=3" and "x^2=x+1" are the same statement. It carries no information beyond how phi is defined.

And landing on a small integer is guaranteed. Every even power does: 3, 7, 18, 47, 123, 322, 843, 2207 — the Lucas numbers. The 3 is L(2), the trace of phi^2 over the rationals. It is not a number phi happens to reach.

So the identity is exact, the proof is sound, phi is pinned down, and none of that is evidence for anything about radix 3. Those are four separate statements and only the first three are established here.