01 · The idea
In everyday language
An exponential-looking Diophantine equation can sometimes be rewritten as a linear combination of logarithms of algebraic numbers. A proposed large solution may force this combination to be extremely close to zero. Baker’s theory gives effective lower bounds for suitable nonzero linear forms, creating a contradiction if the supposed solution is too large. The comparison converts a number-theoretic question into an explicit bound and, in favourable cases, a finite calculation. The key is to rule out impossibly small nonzero quantities, not merely to approximate logarithms on a calculator.
Why it matters
Effective transcendence estimates supplied tools for solving classes of Diophantine equations and deepened the connection between transcendental number theory and explicit arithmetic computation.
