Arithmetic — flashcards

Write a nonzero rational x as p^n · a/b with p dividing neither a nor b; then v_p(x) = n, and v_p(0) = +∞ by convention.

|x|_p = p^−v_p(x) for x ≠ 0, and |0|_p = 0. Divisibility by a high power of p is what it means to be small.

|x + y|_p ≤ max(|x|_p, |y|_p), strictly stronger than the ordinary triangle inequality, with equality whenever |x|_p ≠ |y|_p.

An absolute value on a field satisfies |x| = 0 exactly when x = 0, |xy| = |x||y|, and |x + y| ≤ |x| + |y|. Ostrowski's theorem is the statement that on ℚ these three conditions have far fewer solutions than one would guess.

Every element of ℚ_p is a series in powers of p extending infinitely toward *higher* powers, with only finitely many negative ones — the mirror image of a decimal. In ℚ₅, −1 = 4 + 4·5 + 4·5² + ⋯, which is the honest version of the schoolyard “…999 = −1”.

ℤ_p = { x ∈ ℚ_p : |x|_p ≤ 1 }, equivalently the inverse limit of ℤ/pⁿℤ. A local ring with unique maximal ideal pℤ_p and residue field 𝔽_p.

For positive integers a and b, the remainder sequence is strictly decreasing and non-negative, so it reaches zero after finitely many steps.

For integers a, b there exist integers x, y with ax + by = gcd(a, b). The extended Euclidean algorithm computes them.

The number of division steps in the Euclidean algorithm is at most five times the number of decimal digits of the smaller argument; the worst case is consecutive Fibonacci numbers.

Anthyphairesis on a 1 by √2 rectangle never terminates, which is exactly the statement that √2 is not a ratio of integers.

Every convergent of the continued fraction of x is a best rational approximation of the second kind, and every such approximation is a convergent.

For distinct odd primes p, q the Legendre symbols satisfy (p|q)(q|p) = (−1)^((p−1)/2·(q−1)/2).

For a, b in ℚ*, the product of the local Hilbert symbols over all places equals 1. Reciprocity is this statement, and it is why the completions are not independent.

Every nontrivial absolute value on ℚ is equivalent either to the ordinary absolute value or to a p-adic one for exactly one prime p.

Any consistent, effectively axiomatised theory interpreting arithmetic contains a sentence neither provable nor refutable in it.

Every elliptic curve over ℚ is modular. Proved for semistable curves by Wiles with Taylor (1995), in general by Breuil, Conrad, Diamond and Taylor (2001). Fermat's Last Theorem is a corollary.

If f(a) ≡ 0 mod p and f'(a) ≢ 0 mod p, then f has a unique p-adic root congruent to a. It is Newton's method with the ultrametric inequality doing the work that real-analytic estimates do over ℝ.

A quadratic form over ℚ has a nontrivial rational zero if and only if it has one over ℝ and over every ℚₚ. The first major local–global principle.

For nonzero rational x, the ordinary absolute value times the product of all p-adic ones equals 1. Every place is accounted for, and no single one is privileged.

The sum of the kth powers of the first n integers is a polynomial in n of degree k+1 whose coefficients are the Bernoulli numbers. The defining system is lower triangular and dense: each Bernoulli number depends on every earlier one.

The finite-difference argument that produces a polynomial for positive powers fails for negative ones, because the falling factorials no longer span. The discrete and continuous cases fail for different reasons, which is why the harmonic numbers have an asymptotic expansion but no closed form.

Power sums and elementary symmetric polynomials determine each other by a triangular system. This is how information moves between a polynomial's roots and its coefficients without any root ever being computed.

An entire function of controlled growth is determined by its zeros up to an exponential factor. This — completeness of the zero set plus a growth bound — is what licenses Euler's product for sine, not the fact that the zeros happen to be evenly spaced.

ζ(2) = π²/6, obtained by reading the second coefficient of the sine product as an elementary symmetric function of the reciprocal zeros. The inference runs from coefficients to statistics of the roots, not the other way round.

A Legendre transform applied to a fractional Lagrangian carries the left/right derivative asymmetry from st/fractional-calculus-of-variations into fractional Hamilton's equations: position evolves by a left ('backward-looking') fractional derivative, momentum by a right ('forward-looking') one. The Mittag-Leffler function E_α(z)=∑z^k/Γ(αk+1), the natural fractional generalization of e^z, was verified as an exact eigenfunction of the Caputo fractional derivative (D^α_C E_α(λt^α)=λE_α(λt^α)) by differentiating the defining series term by term, matching to 20 digits. This machinery was applied directly to the Berry–Keating conjecture — H=xp as a candidate classical Hamiltonian for the Hilbert–Pólya program, whose ordinary classical dynamics x(t)=x_0e^t, p(t)=p_0e^{-t} was independently verified (x(3)=20.0855 against predicted x_0e^3=20.0855, H conserved to floating-point precision) — giving fractional Hamilton's-equation solutions in terms of E_α rather than ordinary exponentials, with long-time behavior turning power-law rather than exponential for 0<α<1.

The sum of a function over a lattice equals the sum of its Fourier transform over the dual lattice. It is the bridge back from harmonic analysis to the Bernoulli and zeta material, and the reason the functional equation of ζ looks like a symmetry rather than a coincidence.

An equivalence between categories of local systems on a curve and sheaves on the moduli of bundles over it. Proved 2024 by Gaitsgory, Raskin and collaborators.

Each nonzero residue mod p lifts uniquely to a (p−1)st root of unity in ℤ_p, giving a canonical multiplicative section of ℤ_p× → 𝔽_p×. The residue field sits inside the ring, multiplicatively but not additively.

Nearness in |·|_p and nearness in |·|_∞ are unrelated: by Ostrowski the absolute values are inequivalent, so neither induces any information about the other. Asking whether a p-adic number is larger than a real one has no content.

Because |x + y|_p = max(|x|_p, |y|_p) exactly when the two differ, among any three ultrametric distances the two largest are equal. The metric intuition imported from ℝ fails at the first triangle.

Balls in ℚ_p are simultaneously open and closed, so ℚ_p is totally disconnected — and yet not discrete, since ℤ is dense in ℤ_p.

If c ∈ B(a, r) then B(c, r) = B(a, r). Two balls are therefore either disjoint or nested; partial overlap is impossible.

ℤ_p is a closed subset of the product of the finite discrete spaces ℤ/pⁿℤ, hence compact by Tychonoff — or directly, by pigeonholing digits. ℚ_p is only locally compact.

A series converges in ℚ_p if and only if its terms tend to zero. The condition that is merely necessary over ℝ is sufficient here, and the harmonic series has no p-adic analogue of its failure.

exp_p converges for v_p(x) > 1/(p−1) and log_p(1+x) for v_p(x) ≥ 1; on their common domain they invert one another and give (1 + pℤ_p, ×) ≅ (pℤ_p, +) as topological groups. p = 2 needs the smaller disc.

For odd p, ℤ_p× ≅ μ_{p−1} × (1 + pℤ_p) ≅ ℤ/(p−1)ℤ × ℤ_p — Teichmüller roots of unity times the one-units. For p = 2 it is ℤ/2ℤ × ℤ_2, generated by −1 and 1 + 4ℤ_2.

v_p(n!) = Σ_{i≥1} ⌊n/pⁱ⌋, which grows like n/(p−1). So Σ n! converges p-adically for every prime while diverging in every real sense.

Any field carrying an absolute value has a completion, constructed exactly as ℝ is built from ℚ. This is what puts ℚ_p on the same footing as ℝ rather than treating it as an exotic construction.

If |f(a)|_p < |f′(a)|_p² then f has a unique root α with |α − a|_p < |f′(a)|_p. Dropping the requirement that a be a simple root mod p is what makes the lemma usable in practice.

The zeta function of a variety over a finite field is a rational function — the first of the Weil conjectures, and proved by p-adic analysis rather than by the geometry the conjectures were about.

The Circular Unitary Ensemble (CUE — Haar-random N×N unitary matrices, eigenvalues on the unit circle, joint density ∝ ∏_{j<k}|e^{iθⱼ}−e^{iθₖ}|²) is not an analogy for the deterministic evenly-spaced-points construction used elsewhere on this map (st/wallis-product-three-routes, st/chord-rational-classification) — it is that construction's finite-temperature (β=2) Coulomb-gas generalization, with the deterministic case as its zero-temperature limit. The correspondence to zeta zeros is two separate claims of different strength, chained together: zeta zeros ↔ GUE (the 'GUE hypothesis' — conjectural, though Montgomery proved a restricted form of it, with Odlyzko's numerical verification near height 10^20 as its evidential support) and GUE ↔ CUE (a proven theorem: both converge to the same bulk-scaling-limit determinantal sine-kernel process). The original arc-vs-chord question this resolves: the CUE correlation kernel's denominator, sin((θ−φ)/2), is literally the Chord function already used throughout the Wallis-video material; the GUE-on-a-line kernel is the sine kernel sin(πx)/(πx); the two coincide in the bulk limit because chord and arc agree to leading order as the angular window shrinks — curvature is a finite-N correction, not a persistent distinction. The Keating–Snaith conjecture (moments of |ζ(1/2+it)| match moments of |det(I−U)| for Haar-random U(N)) is the natural next question once the kernels are known to match.

Combining the functional equation ξ(s)=ξ(1−s) with the Schwarz reflection ζ(s̄)=ζ(s)‾ shows every nontrivial zero ρ comes with three siblings: {ρ, 1−ρ, ρ̄, 1−ρ̄}. The critical line Re(s)=1/2 is exactly the fixed-point locus of the involution s↦1−s combined with reflection — so the Riemann Hypothesis is precisely the statement that this quartet always collapses to a pair (ρ=1−ρ̄, i.e. Re(ρ)=1/2), rather than remaining four genuinely distinct points. The Riemann–Siegel Z function was verified in the source conversation to flag a genuine sign change bracketing the first zero, giving a concrete, checkable instance of the general symmetry structure.

For CUE, the maximum of the log-characteristic polynomial obeys max_θ log|P_N(θ)| = log N − (3/4)log log N + O(1) — proved (Arguin–Belius–Harper) — with the −(3/4)log log N correction term the precise signature distinguishing a log-correlated field's extremes from an independent (REM-type) field's, and the whole result an instance of branching Brownian motion universality. The analogous statement for max|ζ(1/2+it)| over a bounded range is conjectured, not proved, by the same mechanism.

The Random Energy Model (Derrida, 1980) — i.i.d. Gaussian energies — has a critical inverse temperature β_c=√(2ln2) separating a quadratic high-temperature free energy from a linear low-temperature (frozen) one, verified by simulation at N=2²² matching both branches (ln2+β²/4 above β_c, β√(ln2) below). The source material names this as sharing its mechanism directly with the Fyodorov–Hiary–Keating conjecture for the maxima of |ζ| on the critical line — already on this map at st/freezing-transition-log-correlated-extremes — making REM the textbook log-correlated-field instance of the exact phenomenon that statement's zeta-zero application borrows.

Katz–Sarnak's theorem for curves over finite fields shows Frobenius eigenvalues — a genuine, proven arithmetic object — converge to classical compact-group statistics, the strongest available non-numerical evidence for the zeta–random-matrix correspondence. Its scope is explicitly bounded: there is no known analogue of Frobenius or étale cohomology over ℤ, so the theorem cannot be strengthened into evidence for the Riemann Hypothesis itself, only for the shape the correspondence takes in a genuinely provable adjacent setting. The logical endpoint of pushing the whole analogy is the Hilbert–Pólya conjecture: the search for a self-adjoint operator whose spectrum is literally the zeros. The CFKRS conjecture — an exact splitting of zeta's moment-asymptotics into a universal random-matrix factor times an arithmetic Euler-product factor — is the structural tool proposed for locating the correspondence's boundary without further numerical computation, though the arithmetic factor itself was named in the source conversation, not computed or explored there.

st/dwork-rationality establishes the zeta function of a variety over a finite field is rational (the first Weil conjecture); the second Weil conjecture (the 'Riemann Hypothesis for curves') states its zeros satisfy |α|=√q. Both were made concrete rather than merely cited: a genus-2 hyperelliptic curve over F_11 was built, its point counts over F_11 and F_121 fed through Newton's identities (st/newton-identities) to obtain the Frobenius characteristic polynomial exactly, and its roots verified to satisfy |α|=√q to eight decimal places. This was then scaled into an actual statistical family — 101 curves at q=11, then 861 at q=31 using a vectorized finite-field-arithmetic implementation — pooling Frobenius angles and tracking the emergence of symplectic-ensemble (USp(4)) endpoint repulsion as q grows, the predicted structure for this family type.

The freezing transition (st/freezing-transition-log-correlated-extremes) produces Gumbel-distributed extreme-value statistics; the Gumbel distribution's skewness constant, 12√6·ζ(3)/π³, was derived here from first principles rather than looked up, by identifying the Gumbel distribution's moment generating function as literally Γ(1−t) — so its cumulant generating function is the same log-Gamma Taylor series whose coefficients are zeta values at integers. This ties Apéry's constant ζ(3)'s appearance in the freezing transition's maximum-value statistics directly to the digamma/zeta coefficient identity already established for the completed zeta function's log-derivative (st/xi-log-derivative-local-global-split).

Analytically continuing the Keating–Snaith formula to complex moment-order β reveals a genuine pole at β=−1/2 — an exact divergence of the formula itself, not merely an asymptotic artifact — traced to the elementary local level-repulsion exponent governing near-coincidence of eigenvalues; confirmed via the argument principle, a small contour around β=−1/2 in the complex moment-order plane having winding number exactly −1. Purely imaginary β=iτ computes something structurally different: the characteristic function of log|det(I−U)|, verified by direct Monte Carlo, with the complete absence of phase-transition behavior on that axis (boundedness by 1 at every τ tested, smooth decay with N) following directly from the fact that a characteristic function of a real random variable cannot diverge.

Kummer's congruences between Bernoulli numbers are the shadow of a single p-adic analytic function interpolating ζ(1−n). A 19th-century family of coincidences turns out to be one object.

The p-adic L-function and the algebraically defined Iwasawa module generate the same ideal. Proved for ℚ in 1984 — worth contrasting with Birch–Swinnerton-Dyer, its classical analogue, which remains open.

Analytic continuation and the functional equation of Hecke L-functions, reproved as Fourier analysis on the adeles. The functional equation becomes Poisson summation.

st/tate-thesis reproves Hecke L-function analytic continuation via Fourier analysis on the adeles. At a single p-adic place, the local piece of that machinery — Tate's local zeta integral Z(f,s) = ∫_{Q_p*} f(x)|x|_p^s d*x — is exactly the p-adic analogue of the (archimedean) Mellin transform, using the multiplicative Haar measure d*x in place of dx/x. For f the indicator of Z_p, it sums to (1−p⁻¹)/(1−p^{-(s+1)}), the local Euler factor itself — so the Euler factor at p is literally a Mellin transform value, not just an analogue of one by loose resemblance.

st/tate-thesis reproves the Hecke L-function functional equation as Poisson summation on Q embedded diagonally in its adele ring A_Q — a consequence of the product formula ∏_v |x|_v = 1 over all places v (finite and infinite), which underlies the whole construction and was checked directly for a specific rational number. Carrying this local/global structure one step further: the log-derivative ξ'(s)/ξ(s) of the completed zeta function decomposes exactly into two pieces — an archimedean piece built from the digamma function (from the Gamma factor at the infinite place) plus a finite-place piece built from the von Mangoldt function (from the Euler product over primes) — verified numerically to 17 digits. The local zeta integral at a single prime p, Z(f,s) for f the indicator of Z_p, sums directly to the Euler factor 1/(1-p^{-s}); the global decomposition is what happens when every place's local computation is assembled at once under the product-formula constraint.

For a polynomial map over Q_p, whether iteration is chaotic is a question about reduction mod p, not about the field being non-archimedean per se. Under good reduction (e.g. f(x)=4x(1-x) at odd p), the multiplier stays a p-adic unit away from the critical point, so |f(x)-f(x')|_p = |x-x'|_p exactly at every step — valuations are preserved, not amplified, and the Julia set is empty (Rivera-Letelier; Benedetto). Under bad reduction (e.g. f(z)=z²+c with |c|_p>1), the identity f(z)-f(w)=(z-w)(z+w) instead gives genuine expansion by |c|_p^{1/2} per step on the Julia set: a repelling fixed point exists there, gaps between nearby points grow by an exact integer factor each iteration, giving a positive Lyapunov exponent and dynamics conjugate to the full 2-shift. So 'chaos is archimedean' is false as a general claim — the real dividing line is good vs. bad reduction, a condition on the map's coefficients relative to p, independent of which absolute value is in play.

Two constructions can give Cantor sets with the same Hausdorff dimension over R while behaving completely differently once the absolute value changes. A digit-restriction set (base-p digits confined to a subset of size d) has dimension log(d)/log(p) at every place where the construction makes sense — purely geometric, place-independent, verified matching a middle-thirds-style real Cantor set exactly. A Julia set's dimension is not: under good reduction (st/padic-reduction-and-chaos) it collapses to 0 in Q_p even where the real Julia set has dimension strictly between 0 and 1, and under bad reduction it takes its own value, log(2)/log(r) for expansion factor r=|c|_p^{1/2}, moving in the opposite direction from the real dimension as a parameter varies — with neither the sum nor the product of the two staying constant, so there is no product-formula analogue for dimension the way there is for absolute values themselves.

In 0 → E(ℚ)/2E(ℚ) → Sel₂(E) → Ш(E)[2] → 0 the middle term is assembled from local conditions at each place and so is mechanically computable; Ш is precisely the local–global failure that remains.

When the Mordell–Weil rank is less than the genus, p-adic integration bounds the number of rational points on a curve. Analysis over ℚ_p answering a question ℝ cannot touch.

For each degree d, Artin's conjecture on forms over ℚ_p holds for all but finitely many p — proved by model theory, by showing ℚ_p and 𝔽_p((t)) become elementarily equivalent as p grows.

A tilting equivalence relating characteristic 0 and characteristic p geometry, which turned a long list of separate p-adic comparison results into instances of one construction.

The reciprocal-square identity π²/sin²(πx) = ∑_{n=−∞}^{∞} 1/(x−n)², obtained by Wästlund's lighthouse argument: equal-brightness sources at every integer, brightness ∝ 1/distance², a circle-doubling construction using the inverse-Pythagorean identity 1/a²+1/b²=1/h² at a right angle, iterated to infinity. x = 1/2 gives ∑ 1/(n−1/2)² = π², which is 8 times the sum of reciprocal squares over the positive odd integers, so that sum is π²/8; since ζ(2) = (odd sum) + ¼ζ(2), this forces ζ(2) = π²/6 — Euler's destination, reached by an elementary route with no complex analysis.

Log-differentiating Euler's sine product sin(πx) = πx·∏(1−x²/n²) once gives the cotangent partial-fraction expansion π cot(πx) = ∑ 1/(x−n); differentiating again gives π²/sin²(πx) = ∑ 1/(x−n)² — exactly the lighthouse formula (st/lighthouse-reciprocal-square). Wallis' product is instead the x=1/2 evaluation of the original, undifferentiated product. So the lighthouse/Basel identity and Wallis' product are two different operations — differentiate twice, versus evaluate at 1/2 — performed on the same underlying product, not two independent facts that happen to coincide.

The Wallis product 2/1·2/3·4/3·4/5·⋯ = π/2 has at least three independent derivations that manage error differently, not three restatements of one argument. (1) Analytic: it is the x=1/2 evaluation of Euler's undifferentiated sine product sin(πx) = πx·∏(1−x²/n²) (st/basel-wallis-same-product) — a substitution into an already-established product, with no limiting process of its own beyond the product's own convergence. (2) Geometric/algebraic: N evenly-spaced points on a circle as the roots of x^N−1, giving two exact finite-N lemmas (observer halfway between two points: distance product = 2; observer replacing one point: distance product = N) whose ratio N·(distance)/2 is an *exact* invariant at every finite N — a single limit, taken once at the very end as N→∞, converts that invariant into π/2 via chord-length-to-arc-length convergence. (3) Classical calculus: Wallis's own route via the reduction formula ∫₀^(π/2) sin^n θ dθ, giving the asymptotic (2n)!!/(2n−1)!! ∼ √(πn) — which stacks two separate approximations (a pointwise limit inside the reduction, and an asymptotic equivalence) that must cancel jointly, rather than carrying one exact quantity to a single final limit. Routes (2) and (3) both pass through the same trigonometric object — a quarter-circle arc — entered through different doors: chord geometry directly, versus a sine substitution turning an algebraic integral into an arc-length integral.

Asking what an irrational f means for the Chord(f) construction splits into two genuinely different questions, not one. (1) The value: for algebraic irrational f, the Gelfond–Schneider theorem characterizes Chord(f)=2sin(fπ) as transcendental. (2) The configuration: N points at an irrational rotation α (rather than N-th roots of unity) obey the three-distance theorem (Steinhaus) — at most three distinct gap lengths between consecutive points on the circle at any N, collapsing to exactly two precisely at the continued-fraction convergents of α. Verified directly for the golden ratio (whose continued fraction is all 1s): exactly two gap lengths at each Fibonacci count N=3,5,8,13,21,34.

For a = 2^m·u, b = 2^n·v with u, v odd 2-adic units, (a,b)₂ = (−1)^(ε(u)ε(v) + mω(v) + nω(u)), where ε(u) = (u−1)/2 mod 2 and ω(u) = (u²−1)/8 mod 2 — both fixed-depth congruences (u mod 4, u mod 8) independent of the size of u.

By Hensel's lemma, whether an odd 2-adic unit u is a square depends only on u mod 8 — a fixed-width digit-read, however large u is. No analogous fixed k exists at an odd prime q: whether u is a quadratic residue mod q is not determined by any bounded number of low bits of u. Checked directly: u=3 vs u=11 (agreeing mod 8) disagree on 339/666 tested primes q; u=7 vs u=71 (agreeing mod 64) disagree on 500/1004 — matching more shared bits does not shrink the disagreement rate toward 0. This is the precise sense in which quadratic reciprocity's supplementary laws are free lookups while the reciprocity step itself is not, and it is what st/hilbert-product's 2-adic term is actually doing.

Iterating n ↦ n/2 for even n and 3n+1 for odd n reaches 1 from every positive integer. Open.

3x³ + 4y³ + 5z³ = 0 is solvable over ℝ and over every ℚₚ, and has no nontrivial rational solution. Found in 1951; it is why the local–global principle is a phenomenon rather than a theorem.

Euclid, c. 300 BC.

Claude Bachet, then Étienne Bézout, 1624 (Bachet), 1779 (Bézout).

Gabriel Lamé, 1844.

the Pythagorean school, c. 450 BC.

Joseph-Louis Lagrange, 1770.

Carl Friedrich Gauss, 1801.

David Hilbert, 1897.

Alexander Ostrowski, 1916.

Kurt Gödel, 1931.

Wiles with Taylor; then Breuil, Conrad, Diamond and Taylor, 1995, completed 2001.

Lothar Collatz, 1937.

Kurt Hensel, 1897.

Hermann Minkowski, then Helmut Hasse, 1890 (Minkowski), 1923 (Hasse).

Ernst Selmer, 1951.

Artin and Whaples, 1945.

Leonhard Euler, 1735.

Leonhard Euler, 1735.

Isaac Newton, c. 1666, published 1707.

Karl Weierstrass, then Jacques Hadamard, 1876 (Weierstrass), 1893 (Hadamard).

Leonhard Euler, 1735.

Siméon Denis Poisson, 1823.

Gaitsgory, Raskin and collaborators, 2024.

Kurt Hensel, 1897.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

József Kürschák, 1912.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Alexander Ostrowski, 1916.

Kurt Hensel, 1897.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

József Kürschák, 1912.

Hasse, Schnirelmann, Teichmüller and others, 1920s–1930s.

Bernard Dwork, 1960.

Kubota and Leopoldt, 1964.

Barry Mazur and Andrew Wiles, 1984.

John Tate, 1950.

Ernst Selmer, 1951.

Claude Chabauty, then Robert Coleman, 1941 (Chabauty), 1985 (Coleman).

James Ax and Simon Kochen, 1965.

Peter Scholze, 2012.

Johan Wästlund, 2010 (Wästlund's paper); popularized 2018 by 3Blue1Brown.

Nat.gcd_eq_gcd_ab — Mathlib.Data.Int.GCD

irrational_sqrt_two — Mathlib.NumberTheory.Real.Irrational

GenContFract.convs — Mathlib.Algebra.ContinuedFractions.Basic

legendreSym.quadratic_reciprocity — Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity

Rat.AbsoluteValue.equiv_real_or_padic — Mathlib.NumberTheory.Ostrowski

padicValRat — Mathlib.NumberTheory.Padics.PadicVal.Basic

padicNorm — Mathlib.NumberTheory.Padics.PadicNorm

padicNorm.nonarchimedean — Mathlib.NumberTheory.Padics.PadicNorm

PadicInt — Mathlib.NumberTheory.Padics.PadicIntegers

IsUltrametricDist.dist_eq_max_of_dist_ne_dist — Mathlib.Topology.MetricSpace.Ultra.Basic

IsUltrametricDist.isClopen_ball — Mathlib.Topology.MetricSpace.Ultra.Basic

IsUltrametricDist.ball_eq_of_mem — Mathlib.Topology.MetricSpace.Ultra.Basic

PadicInt.compactSpace — Mathlib.NumberTheory.Padics.ProperSpace

sub_one_mul_padicValNat_factorial — Mathlib.NumberTheory.Padics.PadicVal.Basic

hensels_lemma — Mathlib.NumberTheory.Padics.Hensel

The continued fraction of x is the quotient sequence of the Euclidean algorithm run on x. Identical work; the CF keeps only the quotients and discards the remainders, which is exactly where the magnitude information lived.

Two representations that each throw away the same thing. Neither lets you read off how big a number is from a prefix of its expansion — the ordering information is simply not in the encoding.

Every Bernoulli number depends on all the earlier ones: the defining system is lower triangular and dense. This is the worst case, and it is the baseline the sparse recurrences get compared against.

The partition recurrence needs only a handful of earlier terms, and the reason is Franklin's involution: almost everything cancels in pairs. Sparse memory is not a lighter problem, it is a heavier theorem — the cancellation has to be proved.

A full orbit determines its seed; orbit statistics do not. Reversing the map is the question of how much of the seed survives being averaged.

Being an integer is what survives every valuation at once — a condition imposed on a larger object, not a starting point that everything else is built up from.

The reals are the completion at the infinite place. Being the ambient number system you happened to grow up with is a fact about pedagogy, not about ℝ.

Power sums are the symmetric functions you get by fixing a particular basis. Newton's identities convert between the choices, which is what makes the choice visible as a choice.

Studying iteration in general and recovering the integer facts as corollaries, rather than building up from the integers and meeting iteration late. A claim about the right order to learn things in, not about what is true.

Symbol manipulation is introduced once arithmetic is automatic.

Counting problems are usually introduced early, alongside arithmetic.

The natural numbers are built from a formal base (e.g. Peano axioms over logic and sets).

Algebraic structures are the axiomatic generalisation of arithmetic's operations.

Formulas and proofs ↔ natural numbers, so that provability becomes an arithmetic relation. This is what lets arithmetic talk about itself.

Abelian extensions of a number field ↔ quotients of its idele class group. The abelian, fully proved fragment of Langlands.

Quadratic reciprocity is the first visible instance of the pattern class field theory completes for all abelian extensions.

Galois representations ↔ automorphic representations, matched by their L-functions. Proved for elliptic curves over ℚ (modularity, Wiles–Taylor, giving Fermat) and in many other cases; open in general.

Recorded as conjectural: expected, not proved.

Each Galois representation produces an L-function — the arithmetic side's contribution to the shared currency.

Each automorphic form produces an L-function too. Langlands asserts these two supplies of L-functions coincide.

Reciprocity is the statement that the local Hilbert symbols over every completion multiply to 1. So the p-adic worlds are not independent — fixing all but one determines the last. This was your own question: whether quadratic reciprocity means the completions are coupled. It does.

Pair correlation of the zeta zeros ↔ eigenvalue statistics of large random Hermitian matrices. Numerically overwhelming, still conjectural.

Recorded as conjectural: expected, not proved.

Ostrowski's theorem says the branches are exactly ℝ and one ℚₚ per prime. This is the cleanest fork in mathematics: infinitely many consistent completions of the same starting object, and no principled reason to privilege the real one.

Ordinary analysis is what you get by taking the one branch we happen to live on. Naming it a branch rather than the default is the whole point of the lens.

The alphabet size is the number of inverse branches of the map. Choosing base p and choosing to work p-adically are versions of the same choice, differing in how the refinement is oriented — disjoint balls rather than intervals sharing boundaries.

You may study a problem at a single completion, or assemble every place at once into the adeles. Neither is the default; the product formula says no place is privileged, and choosing one is a choice.

An apparent fork that collapsed. Letting the root vary at each stage looks like a new family, but every canonical version is continued fractions conjugated by a logarithm — and the base of that logarithm turns out to be irrelevant, even numerically. Testing whether a fork is real is half the work of this lens.

Recorded as apparent: the resemblance is real, the mechanism is not shared.

A continued fraction is the run-length encoding of a geodesic — the forward itinerary of the Gauss map. The digits stop being digits and become a record of where a point went.

Forwards, the fate of a Collatz orbit is undecidable in practice. Backwards, the orbit hones in on its seed — and the question of how fast becomes a question about linear forms in logarithms, which is where Baker's theorem lives.

From below, succession is an axiom and addition is defined from it. From above, addition is one rung of the hyperoperation ladder and the Peano axioms read as theorems about the bottom of it. Same content, opposite direction of justification.

Euclid's algorithm consumes a pair and reduces it; the continued fraction it produces generates approximations outward. The same recursion read as destruction or as construction depending on which end you stand at.

Moving the recursion up one hyperoperation level buys expressive power and costs base-independence — the property that made continued fractions worth having.

Proposed cost that was not one. Both discard something about distance, but not the same something: p-adic expansions are positional and linear, continued fractions relative and recursive. The refutation forced a distinction worth more than the conjecture.

Recorded as refuted: this edge was drawn and then withdrawn.

Hasse–Minkowski gives local-implies-global for quadratic forms. Raise the degree by one and Selmer's cubic destroys it. The cost of a single extra degree is the entire principle — and the Tate–Shafarevich group exists to measure exactly how much was lost.

Not a theorem — an instance of the same shape. The interest of Collatz here is structural: a rule this simple should not be able to hide its own outcome, and it does.

Tested directly: nested radicals do have a series dual, via difference-of-squares telescoping, echoing continued fractions' own series representation. The pairing is real, but it isn't the same shape — it comes out all-positive rather than alternating, which is exactly what breaks optimality for nested radicals. A confirmed analogy that differs in a load-bearing way from the thing it was tested against.

Partial. No direct interpretation the way binomial coefficients count something outright, but one step removed — through the Genocchi and zigzag numbers, both of which do have direct combinatorial meanings. A correspondence that holds once you're willing to go via an intermediate object.

Recorded as conjectural: expected, not proved.

Held, and closed a loop rather than opening a new one: Poisson summation is the bridge back from Fourier analysis to the Bernoulli/zeta material, and it's the reason the functional equation of ζ looks like a symmetry instead of a coincidence.

REFUTED at the level asked. Checked against Maynard's actual definition of an admissible tuple, and the objects are not the same. A real connection was found one level up instead — at the level of technique (moment methods), not of the objects themselves. The surface-level naming match was the wrong place to look.

Recorded as refuted: this edge was drawn and then withdrawn.

`Mathlib.NumberTheory.Ostrowski` imports `Mathlib.Analysis.AbsoluteValue.Equivalence`, `Mathlib.Analysis.SpecialFunctions.Log.Base`, `Mathlib.Analysis.SpecialFunctions.Pow.Continuity`. Read off Mathlib/NumberTheory/Ostrowski.html. No human asserted this edge.

`Mathlib.NumberTheory.Ostrowski` imports `Mathlib.NumberTheory.Padics.PadicNorm`. Read off Mathlib/NumberTheory/Ostrowski.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.PadicVal.Basic` imports `Mathlib.NumberTheory.Divisors`. Read off Mathlib/NumberTheory/Padics/PadicVal/Basic.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.PadicVal.Basic` imports `Mathlib.Data.Nat.MaxPowDiv`, `Mathlib.Data.Nat.Multiplicity`, `Mathlib.Data.Nat.Prime.Int`. Read off Mathlib/NumberTheory/Padics/PadicVal/Basic.html. No human asserted this edge.

`Mathlib.Data.Int.GCD` imports `Mathlib.Algebra.Divisibility.Basic`, `Mathlib.Algebra.GroupWithZero.Semiconj`, `Mathlib.Algebra.GroupWithZero.Divisibility`, `Mathlib.Algebra.GroupWithZero.Nat`. Read off Mathlib/Data/Int/GCD.html. No human asserted this edge.

`Mathlib.Data.Int.GCD` imports `Mathlib.Algebra.Group.Commute.Units`, `Mathlib.Algebra.Group.Int.Defs`, `Mathlib.Algebra.Group.Nat.Defs`, `Mathlib.Algebra.Group.Nat.Units`. Read off Mathlib/Data/Int/GCD.html. No human asserted this edge.

`Mathlib.Algebra.ContinuedFractions.Basic` imports `Mathlib.Algebra.Field.Defs`. Read off Mathlib/Algebra/ContinuedFractions/Basic.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Algebra.Polynomial.Identities`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Analysis.SpecificLimits.Basic`, `Mathlib.Analysis.Normed.Field.ProperSpace`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

`Mathlib.NumberTheory.Padics.Hensel` imports `Mathlib.Topology.Algebra.Polynomial`, `Mathlib.Topology.MetricSpace.CauSeqFilter`. Read off Mathlib/NumberTheory/Padics/Hensel.html. No human asserted this edge.

The natural numbers axiomatised: zero, successor, and induction. The base object almost everything else is eventually built on or compared to.

Repeated division to find the gcd of two numbers — the oldest algorithm still in everyday use.

x² − ny² = 1, solved via continued fractions. The classical entry point where number theory starts reaching into algebra, and where continued fractions stop being a curiosity.

Whether p is a square mod q is determined by whether q is a square mod p. Gauss called it the golden theorem; it is the seed of the whole reciprocity story.

The Riemann zeta function encodes the primes analytically; its zeros control their distribution.

The abelian case of reciprocity, completed in the 1920s–30s: it classifies all abelian extensions of a number field in terms of data inside the field itself.

Actions of a field's absolute Galois group on vector spaces. The arithmetic side of the Langlands correspondence.

Solve a problem over the reals and over every ℚₚ, and ask whether that forces a solution over ℚ. It holds for quadratic forms and fails for cubics — the failure is where the interesting objects live.

All completions of a number field handled at once, as a single restricted product. The device that turns ‘check every place separately’ into one object with its own analysis.

Class groups studied along infinite towers of number fields, where their growth becomes regular and is captured by a p-adic L-function.

Analytic invariants attached to arithmetic and to automorphic objects alike. They are the common currency in which both sides of Langlands are compared.

For each prime p, a completion of the rationals in which closeness means divisibility by high powers of p. Non-Archimedean: every triangle is isosceles, and every point of a disc is its centre.

Every absolute value on ℚ is equivalent to the ordinary one or to a p-adic one. This is the fork itself: it says exactly what the branches are and that there are no others.

Write a number as an integer plus one over an integer plus one over… Unlike decimal it has no fixed alphabet, which is what buys it base-independence and best-approximation.

Succession, addition, multiplication, exponentiation, tetration — each level iterating the one below. Cross-confirmed: this ladder appears in both transcripts as a place to move up and down.

Halve if even, 3n+1 if odd. Whether every start returns to 1 is open. Its interest here is structural: the fate of an orbit cannot be determined in advance.

Algebraic integers whose conjugates all sit inside the unit circle. The plastic number is the smallest; nothing lives between 1 and it — provided you require an algebraic integer, which is the sloppiness you caught.

Effective lower bounds on linear forms in logarithms. Where reverse-orbit reasoning about Collatz-like maps actually bottoms out.

The coefficients that turn a sum of kth powers into a polynomial in n. The defining system is lower triangular and dense — every Bernoulli number depends on all the earlier ones — which makes them the standard against which sparser recurrences get measured.

Good at — Exact facts about individual numbers. Is this prime, does this equation have integer solutions, what exactly is the greatest common divisor. Answers here are yes or no, with no error bar.

Bad at — Anything approximate or continuous — it has almost no native vocabulary for “roughly”. It is also famously unable to predict its own difficulty: one-line questions here stay open for centuries, and nothing internal to the area explains why.