The Basel Problem asks for the exact value of the infinite sum 1 + 1/4 + 1/9 + 1/16 + ... — the sum of the reciprocals of all perfect squares. Posed in 1650 and resisting the finest mathematicians of the era for over 80 years, it was finally solved by a 28-year-old Leonhard Euler in 1734, who found the strikingly unexpected answer π²/6 — connecting the circle constant π to a purely number-theoretic sum with no obvious circular geometry in sight.
Add up the reciprocals of the perfect squares, forever: 1/1 + 1/4 + 1/9 + 1/16 + 1/25 + .... This sum obviously converges to some finite number (see the Series & Convergence entry) — but to exactly what number? This innocent-looking question, first posed by Pietro Mengoli in 1650, stumped Europe's greatest mathematicians — including several members of the famous Bernoulli family — for over 80 years.
Euler's genius insight was to treat the sine function — a function intimately tied to circles and periodic motion — as if it were an "infinite polynomial," and to compare its known series expansion (see the Series & Convergence entry) against the equivalent product form built from its roots. Since sin(x)'s roots are located at multiples of π, this comparison directly smuggled π into an answer about an ostensibly unrelated sum of reciprocal squares — an audacious and, at the time, not fully rigorously justified piece of mathematical reasoning that nonetheless gave the correct answer.
The problem is named after Basel, Switzerland — the hometown of both the extended Bernoulli family (who worked on it extensively without success) and Euler himself, who finally solved it there.
Euler's Taylor series for sine (see the Derivative entry) is:
Euler then boldly treated this as though it were literally an infinite polynomial, reasoning that — just as an ordinary polynomial can be factored using its roots — this "polynomial" should also factor using the known roots of sin(x)/x, namely x = ±π, ±2π, ±3π, ....
Writing out the corresponding infinite product form of sin(x)/x based on these roots, and then comparing the coefficient of x² on both sides of the resulting equation (the Taylor series version versus the infinite product version), Euler derived:
Multiplying both sides by π² and rearranging gives exactly π²/6 = 1 + 1/4 + 1/9 + 1/16 + ... — the celebrated result.
Euler's original 1734 argument, while producing the correct answer, relied on treating an infinite series as though ordinary polynomial factorisation rules applied directly to it — a step that was not rigorously justified by the mathematical standards of Euler's own era, let alone by later, much stricter 19th-century standards of analytical rigour. Euler himself later (1741) supplied a more careful and fully rigorous proof, and numerous alternative, fully rigorous proofs (using Fourier series, complex analysis, and other methods) have since been developed.
π²/6 ≈ 1.644934.... Adding up the first several terms of the original series (1 + 0.25 + 0.111 + 0.0625 + 0.04 + ...) does indeed slowly approach this value, though convergence is fairly slow — reflecting the general difficulty of numerically verifying such results to high precision using the original series form directly.
The Basel Problem is precisely the statement that ζ(2) = π²/6, where ζ is the Riemann zeta function (see the Riemann Hypothesis entry), defined by ζ(s) = Σ 1/nˢ. Euler didn't stop at s=2: he also computed ζ(4) = π⁴/90, ζ(6) = π⁶/945, and in general found an exact closed-form expression for ζ(2n) for every positive integer n, all expressible in terms of π^(2n) and specific rational numbers called Bernoulli numbers.
Remarkably, and in striking contrast to the clean, fully general formula for even integers, essentially nothing comparably clean is known about ζ(3), ζ(5), ζ(7), and other odd-integer values of the zeta function. Roger Apéry proved in 1978 (in a genuinely surprising and celebrated result, initially met with considerable scepticism from other mathematicians until the proof was thoroughly checked) that ζ(3) is irrational — a value now called Apéry's constant — but no simple closed-form expression analogous to the even-integer case is known, or even believed to exist, for ζ(3) or any other odd zeta value. Whether ζ(5) is irrational remains formally unproven, though widely believed to be true, as of 2026.
Beyond Euler's original heuristic argument, fully rigorous proofs of the Basel Problem include: a proof via double integration (elegantly evaluating a specific double integral two different ways), a proof via Fourier series (expanding a simple function and applying Parseval's theorem), and a proof via complex analysis (using the residue theorem — see the Complex Number entry). The existence of so many genuinely different valid proof techniques, each illuminating a different facet of why the result is true, is itself considered a notable and celebrated feature of this particular theorem within the mathematical community.
A striking and unexpected application: the probability that two randomly chosen positive integers share no common factor (are coprime) is exactly 1/ζ(2) = 6/π² ≈ 0.6079 — meaning roughly 60.79% of randomly paired integers are coprime. This elegant connection between an abstract infinite sum and an intuitive probabilistic question about ordinary whole numbers is a favourite illustration of the deep, often surprising unity connecting seemingly disparate areas of mathematics.