Irrationality of Catalan’s constant

The explanation

Catalan's constant is the sum 1 − 1/9 + 1/25 − 1/49 + … and so on forever. The bottom numbers are the odd numbers squared, and the signs alternate. Call the total G. It is about 0.9159656. Stop after four terms and you get the fraction 10016/11025, about 0.90848. That misses G by about 0.0075. More terms get closer. The question this paper settles is whether G itself is a fraction.

Closeness alone cannot answer that. Suppose G were a fraction a/b. Pick any whole numbers A and B. Then A times G minus B is a whole number of b-ths. So it is either zero or at least 1/b in size. To show G is not a fraction, it is enough to find whole numbers A and B with one property. A times G minus B must never be zero, yet must shrink toward zero. A number that is not a fraction is called irrational.

The obvious attempt fails. Take A = 11025 and B = 10016 from the four-term sum. Then A times G minus B is about 82.5. With six terms the bottom number is 12006225, and the same quantity is about 40865. The fractions improve, but multiplying by the bottom number wipes out the gain. The paper calls this the basic difficultySource 1. Faster recipes for G exist. In 2003, Wadim Zudilin built several. He noted that their whole-number combinations still do not tend to zero. Roger Apéry's 1979 method for two related sums worked because the shrinking survived that multiplication. Similar recipes for G had not managed that.

Earlier results came at G from the side. G is the first of a family of sums, β(2), β(4), β(6) and so on, one per even power. In each, the odd numbers are raised to that power instead of squared. In 2003, Tanguy Rivoal and Zudilin proved that at least one of β(2) through β(14) is irrational. Zudilin cut that list to six values in 2019. Li Lai and Li Zhou cut it to five, through β(10), in 2022. Such results promise an irrational member without saying which one.

This paper proves that G is irrationalSource 2. It replaces the single combination with large square grids of numbers. A grid of the paper's chosen size 48N is built for each whole number NSource 3. Each entry mixes 1, G and a second constant, ζ(2) = 1 + 1/4 + 1/9 + 1/16 + …Source 4. The rows are chosen so the ζ(2) parts cancel in every entry. Each grid then gets one number computed from all its entries, called its determinant. If G were a fraction, every determinant would be a fraction tooSource 5.

Now the trap from before works on a larger scale. Control of the bottom numbers stops a nonzero determinant from being very small. An integral formula for the same determinant forces it to be smallSource 6. After a fixed rescaling, the first bound stays above −2.29084. The second stays at or below −2.290939875Source 7. They clash by about one ten-thousandth. That is a thin margin, but any clash is a contradiction. Zudilin's 2017 determinant method is the paper's closest precedentSource 8. He explained why the approximations available to him could not apply it to G.

The hard part is that the determinant must not be zero, because a zero springs no trap. The paper handles this when N is a large prime number. Assuming G is a fraction, each growing grid reduces to three fixed grids of fractions. The paper claims an exact finite computation shows those are not zeroSource 9. That yields nonzero determinants for an endless run of primes, which is all the trap needsSource 10.

The result says only that G is not a fraction. It gives no irrationality measure, a limit on how closely fractions can approach GSource 11. It does give a side result. Ian Agol showed that 4G is the smallest volume of a certain kind of curved three-dimensional space. That space has exactly two cusps, ends that taper off forever. So that smallest volume is irrational tooSource 12. Zhi-Wei Sun announced a proof of the same statement in a September 2026 preprint. This paper uses nothing from itSource 13.

How sure is this? A computer proof checker called Lean has a formal proof that G is irrational. A tool called Comparator matches it to a fixed reference statement. The step about volumes is not in the Lean scope note. No specialists have published a review yet.

What it doesn't show. This does not say how closely fractions can approach Catalan's constant.