Irrationality results and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Irrationality from small linear forms — Lemma 1.1. A real number x admitting integers a_n, b_n with a_n + b_n x nonzero for every n and a_n + b_n x -> 0 is irrational.
• Size of lcm(1, ..., n) — Lemma 1.2. log lcm(1, ..., n) = psi(n); by the prime number theorem lcm(1, ..., n) <= e^((1+eps)n) for every eps > 0 and all large n.
• Monomial integrals — Proposition 2.3. Integers r, s >= 0: the integrals of x^r y^s/(1-xy) and -log(xy) x^r y^s/(1-xy) over the unit square are zeta(2) - sum_{k<=r} k^-2 and 2(zeta(3) - sum_{k<=r} k^-3) if r = s, and explicit harmonic-type rationals with denominators dividing lcm(1, ..., max(r,s))^2 and ^3 if r != s.
• Irrationality of zeta(2) — Theorem 4.2. zeta(2) = pi^2/6 is irrational; consequently pi^2 is irrational.
• Apéry's theorem — Theorem 5.3. zeta(3) = sum k^-3 is irrational.
• Chebyshev rows — Proposition 3.1. The rows P_r and D_r, built from Chebyshev polynomials T_d and U_(d-1), are integer polynomials with monomials in degrees A to H-1 (respectively A+1 to H-1), and D_g = 0.
• Taylor contact — Lemma 3.2. As power series in t, tP_r/f - D_r = O(t^L).
• Starting values 4G and (3/2)zeta(2) — Lemma 5.3. The arcsine-weighted mixed moments satisfy M(0,0) = 4G and M(0,1) = pi^2/4 = (3/2) zeta(2); in particular zeta(2) = pi^2/6.
• Decomposition of the moments — Proposition 5.4. M(i,j) = M0(i,j) + 4G c_((i-j)/2) + (3/2)zeta(2) c_((j-i-1)/2) and Z(i,j) = Z0(i,j) + [i=j] zeta(2), with rational arrays M0, Z0 and Catalan numbers c.
• Rationality of the determinants — Proposition 6.1. Every raw entry of the determinants lies in Q + QG; if Catalan's constant G is rational, every determinant Delta_N is rational.
• Transfer of minor bounds — Lemma 1.4. Matrices over Q_p whose full minors have valuation at least -Theta: the bound persists for det(FT), T over Z_p, and for FU^-1, U over Z_p with unit determinant.
• Denominators at the prime 2 — Proposition 4.1. If G is rational: v_2(Delta_N) >= -(505/4608) n^2 - O_G(n log(n+2)), n = 48N, and the same holds for every full minor of the raw matrix.
• Base-p digit reductions — Proposition 3.1. Odd primes p: p^2 M0(i,j) is congruent modulo p to E_d M0(i',j') and p^2 Z0(i,j) to Z0(i div p, j div p) or 0, with explicit base-p digit indices i', j', d.
• Leading layer — Proposition 4.1. Every raw column lies in p^-2 Z_(p)^n, with an explicit reduction of p^2 F_j modulo p in terms of the arrays at index j div p.
• Finite-place lower bound — Proposition 6.1. If G is rational: along every sequence with Delta_N != 0, liminf (log|Delta_N|/n^2 - (1/2) log 2) >= -8609/4608 - (1/2 + 505/4608) log 2 > -2.29084.
• Nonvanishing at prime scales — Proposition 1.1. If G is rational: for every sufficiently large prime p, v_p(Delta_p) = -96p; in particular Delta_p != 0.
• The fixed certificate — Proposition 1.2. Three explicit rational 48 x 48 determinants, det B_0 and det(B_0 +- B_1), are nonzero; proved by elimination modulo 101.
• Andréief's identity — Lemma 1.1. The integral of det[f_r(y_i)] det[g_k(y_i)] against mu^n equals n! det[integral of f_r g_k d mu] whenever the integrand is integrable.
• Integral formula for the determinants — Proposition 2.1. Delta_N = (n!)^-2 times an integral over (-1,1)^n x (0,1)^n of a mixed two-branch determinant against Vandermonde and Cauchy factors.
• Bounded interpolation — Lemma 3.1. Distinct nonzero points x_i in (-1,1) with sum (1-x_i^2)/(1+x_i^2) <= D: a function holomorphic near the closed unit disk takes the values c_i x_i^D (c_i = 1 or -5 by the sign of x_i) and is bounded by 10 e^12 n.
• Reduction to a supremum — Proposition 6.1. |Delta_N| <= 2^n K_n (n!)^-2 max over two cases of the supremum of explicit majorants; hence log|Delta_N|/n^2 - (1/2) log 2 is at most their normalized supremum plus O(log(n+2)/n).
• Energy reduction — Proposition 3.2. For any two trial coefficient sequences, the normalized majorant is bounded by an explicit constant plus the suprema of two functions of one variable.
• Descartes' rule of signs — Lemma 4.1. A nonzero real polynomial has at most as many positive roots, counted with multiplicity, as sign changes in its coefficient sequence.
• Real-place upper bound — Proposition 4.3. Unconditionally, limsup (log|Delta_N|/n^2 - (1/2) log 2) <= -2.290939875 < -2.2909.
• Irrationality of Catalan's constant — Theorem 5.1. Catalan's constant G = sum_(j>=0) (-1)^j (2j+1)^-2 is irrational (OpenAI, 2026).

Reading order

Read the first lesson for the method of small linear forms, Beukers' integrals and the irrationality of ζ(2) and ζ(3). The second lesson builds the determinants for Catalan's constant and proves that they would be rational if G were. The third and fourth lessons bound their denominators at the prime 2 and at the odd primes, the fifth proves that they do not vanish at prime scales, and the last two bound them from above at the real place and conclude.