Two Searches That Would Break Computability

Imagine running two procedures side by side. Search A lists integer tuples and checks whether f equals 0.

#computability #hilbertstenth #undecidability #numbertheory

Why Linear and Quadratic Growth Can’t Coexist

A line and a parabola can intersect, so the animation is not claiming that their shapes alone prove anything.

#ellipticcurves #numbertheory #hilbertstenth #mathvisualization

Why Defining Integers Inside Rationals Is So Hard

The equation 2x − 1 = 0 exposes the gap immediately: x = 1/2 is rational, but it is not an integer. So undecidability over integers does not automatically transfer to rational solutions.

#hilbertstenth #rationalnumbers #logic #numbertheory

Why Integer Zeros Fit Through a Finite Corridor

A bounded search over integers really can become a finite checklist. If a problem supplies a bound B, then any relevant integer zero of f(n) = 0 must be among the values from −B to B.

#hilbertstenth #integers #rationalnumbers #computability

Why Rational Solutions Change Hilbert’s Tenth Problem

An equation can have integer roots, rational-but-not-integer roots, or no rational roots at all. For x² − 4 = 0, the solutions are −2 and 2, so integers work.

#hilbertstenth #rationalnumbers #undecidability #numbertheory

Why Finding a Root Is Easier Than Proving None Exists

A search can certify a yes answer without ever certifying a no. List integer tuples one by one and test a polynomial at each: if one makes the value 0, the search eventually finds it.

#undecidability #haltingproblem #hilbertstenth #numbertheory

How Elliptic Curves Could Encode Integers Inside Rationals

Follow for the equations hiding behind the news.

#ellipticcurves #numbertheory #undecidability #hilbertstenth