Number Theory
Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
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The result settles an Erdős-Straus conjecture, proves a series involving omega irrational, and gives asymptotics for equal-omega consecutive
· “Quantitative correlations and some problems on prime factors of consecutive integers”
For short intervals of length x^θ, the new relation gives μ(17/30) ≤ 7/12 and μ(2/15+Δ) ≤ 1−(9/13)Δ.
· “On the number of exceptional intervals to the prime number theorem in short intervals”
The supremum of orthogonality defects of HKZ-reduced bases satisfies log D_n / (n log n) -> 2 as n -> infinity, and D_4 = 4375/1024.
A block condition on the inner word completely decides the equality cases of the deconvolved measure.
· “The equality cases P_t(mathbb{N})=tfrac12 for the deconvolved sum-of-digits measures”
A closed ray of Schwartz functions yields the 196560 ratio through Poisson summation.
· “The 196560 auxiliary-function conjecture for the Leech lattice”
Even entire functions keep the quotient pencil real-rooted up to this bound; one step beyond, degrees 3 and 5 fail.
· “Jacobi Endpoint Pencils and Sharp Interlacing for Centered Binomial Samples”
The exact reducibility criterion leaves no room for digraphs one vertex short of the Moore bound.
· “A proof of the cyclotomic conjecture and the non-existence of almost Moore digraphs”
A flow cross-section with finite measure makes face statistics universal for almost every sail.
· “Generalised Gauss--Kuzmin Distribution for Klein sails in mathbb R³”
The characterization links dualizable additive categories to idempotent ideals, nuclear modules, and algebraic K-theory.
The (1/3,1/3) arithmetic-progression case of the Strong Divisor Conjecture is false, even with sub-polynomial slack.
· “Refuting a Conjecture of Umans and Wang on Arithmetic-Progression Divisor Covers”
Each p-primary piece is a small explicit tree family; only one shape never occurs, and the rest are all realizable.
· “Isogeny graphs of elliptic curves in characteristic zero”
Matching lower and upper bounds settle Erdős Problem 768 and bound simple-group orders
· “A Resolution of ErdH{o}s Problem 768: the Sylow Divisor Condition”
The count of pairs satisfying σ(a)+σ(b)=σ(a+b) with a+b≤x diverges faster than linear at every logarithmic scale.
· “A resolution of ErdH{o}s Problem 1061 on the sum-of-divisors function”
The quadratic coefficient a must meet a prime-dependent valuation threshold at every p^α exactly dividing N.
· “A Local Valuation Criterion for Quadratic-Permutation Interleaved Zadoff--Chu Sequences”
This resolves the final open case in the metric theory of inhomogeneous Diophantine approximation and aligns it with the homogeneous theory.
A new anima with the Weil group as fundamental group adds higher homotopy to produce better-behaved cohomology.
Density-1/2 sets satisfy |A Δ (ax+b)| = o(p) for all |a|,|b| = o(log p), solving Green's open problem 90.
· “Almost Affine Invariance Over Prime Fields: Green Problem 90”
Generating function with odd smallest part matches conjectured analytic sum via indefinite theta functions.
· “Proof of a conjecture of Andrews and Bachraoui on a Hecke sum”
For every admissible ratio (an+b)/(an+d), values either cycle through µ_k or fill the whole circle.
· “Value distribution of multiplicative functions along linear fractional sequences”
Delta Frobenius on jet spaces gives Hodge-Pink structures, recovering the Tate module for the Carlitz case.
· “Delta theory of Anderson Modules II: Hodge-Pink structure”
The count follows the prime-factor anatomy of n, with phase transitions at a critical ratio and one open regime.
· “Integers divisible by a shifted prime in a given interval”
A sharper distribution level lifts the earlier P8 result by one sieve step, keeping the expected x/(log x)^2 count.
For even prime powers, the share of primitive elements determines when a pair of adjacent non-primitive elements must exist.
· “On Consecutive Non-primitive Elements over Finite Fields”
Over F_q(t) with large constant field and characteristic above the degree, a smooth degree-d hypersurface satisfies the circle-method…
· “Birch's theorem over function fields with quadratically many variables”
Every solution of X^2+Y^2=Z^2 in GL2(Z), up to signs and simultaneous conjugation, falls into one of two isolated orbits plus one family…
Under a refined Gan–Gross–Prasad conjecture, degree-2 Siegel norm ratios become explicit weighted sums of central L-values.
If Birch–Swinnerton-Dyer is true, counting rational points reduces to an L-function evaluation.
· “Introduction to the Birch and Swinnerton-Dyer Conjecture”
For each Hamming weight, the exact floor on the sum-of-digits bias is now known, not just an exponential estimate.
· “Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight”
Under Selberg's conjecture, averaged squares of mixed L-values are determined up to a genuine power-saving error.
· “The second moment of twisted modular L-functions and Dirichlet L-functions at the central point”
Each output bit of the Collatz step becomes a finite formula in earlier bits.
· “A Boolean polynomial operator for the Collatz 3n+1 problem”
A free spectral parameter yields central values, central derivatives, and rank-one results.
· “Fourier Spectral Reciprocity and Canonical Hecke L-Functions”
An Atkin–Lehner twist makes Poincaré duality pair cohomology classes in p-adic families, recovering the modular-curve case.
A one-parameter identity reduces fifty dilogarithm terms to five, reaching totally real quartic ladders.
· “Collapsing Fifty Dilogarithm Arguments to Five Terms over mathbb{Q}bigl(u,sqrt{4-3u²}bigr)”
A new regulator bound for cubic Thue–Mahler equations cuts the previous exponent on |k| from 1+ε to 1/2.
· “Power-saving bounds for Thue--Mahler and Mordell equations”
The count matches the trivial lower bound up to a boundary-order error, settling the Elsholtz–Rackham conjecture.
For p-adic Lie groups, every central symmetry of compact mod-p representations comes from the group itself, up to nilpotents.
An elementary proof rules out triples whose middle exponent uses only primes congruent to 5 mod 8, extending the cube case.
· “An elementary note on three consecutive powerful numbers”
Lacunary sampling of nearest-neighbour distances yields explicit Weibull, Fréchet, and logarithm laws.
· “Extreme values and logarithm laws for toral translations”
Any sequence of $p^3+3p-3$ elements contains a product-one subsequence of exactly $p^3$ terms.
A deterministic sin^2-type algorithm returns certified unit cycles on every tested complex cubic field.
A single volume bound, plus continued fractions, classifies norm-Euclidean quadratic, cubic, and quartic rings.
Tribonacci unsigned representation counts become a product polynomial, with a Bernoulli-convolution limit.
For smooth curves over Cp, these intersections concentrate at the unique canonical point, except in a supersingular exceptional case.
· “p-adic Equidistribution of Special Loci in a Product of Modular Curves”
Finitely many powers of 3 have a binary digit word of small factor complexity, such as a Sturmian word.
A new unconditional average links the Hardy–Littlewood integral to Fermat's Last Theorem
For two transversal smooth conics over a finite field, the exact numbers of points in the four internal/external position classes are…
· “An Exact Counting Formula for the Mutual Position of Two Plane Conics”
For every fixed width m, the rectangle partition number p(m,n) satisfies log p(m,n) = π√(2mH_m/3)√n + O(log n), settling the conjecture.
· “The asymptotic behavior of the rectangle partition function p(m,n)”
For Zariski-generic coefficient vectors, hyperelliptic families y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t are generically ordinary at every large…
· “Generically Ordinary One-parameter Hyperelliptic Families are Dense”
The paper derives cyclotomic norm congruences for p-adic zeta and modular L-values, but the GL1 proof appears to use the wrong Teichmuller…
· “Cyclotomic norm congruences for zeta values and modular L-values”
The first moment has complete secondary term q^{2g/3}(alpha_{g mod 3} g + beta_{g mod 3}) with coefficients of minimal period exactly 3…
· “The Period-Three Secondary Term in the Mean Value of L(1/2,chi_D) in the Hyperelliptic Ensemble”
Infinitely many Diophantine triples get an integral point where no factor is a square.
The paper reduces uniqueness of the main conjecture to SK_1 vanishing on division algebras and proves low-type cases.
High-dimensional lattice packings are now provably denser than previously certified.
· “On the constant in Klartag's theorem for lattice packings in high dimensions”
A new theorem lists the exact digit pattern of every such pair and rules out all others.
A recursive coset-splitting construction makes every line direction a limit point while keeping crossings off the lattice.
· “On the directions occurring in lattice-line coverings of the integer plane”
For every derived vector bundle over a derived Artin stack, the motive-based transform inverts itself up to a shift.
A local p-adic analogue of half-integral-weight newforms exists exactly for generic representations, with dimension 1 or 2.
Square-class quotients of Z/nZ are classified; triviality, hyperbolicity, and the invertible part all have exact answers.
Local lattice-counting and overorder series coincide, with uniform Kloosterman factors as payoff.
· “Yun's zeta function and the overorder zeta function for Gorenstein cubic orders”
The generic normalized value of the new approximation function N(α) is zero, yet integers keep exact positive values.
· “The normalized approximation function for multiple zeta-star values”
For |n|>1, the Tonelli–Shanks power satisfies no polynomial identity modulo almost all primes.
· “Transcendence of Tonelli--Shanks Power Modulo Infinitely Large Primes”
Implicit mutual authentication, 192-byte sessions, and a single tag check that rejects forged messages before any isogeny is computed.
· “MCSI: A Masked Commutative Supersingular Isogeny Key Exchange with Blinded Ephemeral Keys”
A twisted Möbius estimate over finite fields would prove exceptional Galois polynomials are genuinely rarer.
· “The strong form of Van der Waerden's conjecture via twisted Chowla”
One finite Gaussian identity then yields Kummer's carry rule, mod-5 congruences, and overpartitions.
A short argument upgrades the old one-value result to two, one of them independent of 1 and ζ(3).
Complete classifications for sizes 7–9 and the first multiplicative-subgroup MSTD sets complete the small-case picture.
A real smallest zero predicts linear spectral plots; a complex pair predicts oscillation — in 30 of 30 unambiguous cases.
The result leaves a single unresolved square-free order, 110, where the one-step lifting argument is provably blocked.
· “Prime multipliers of order ten: norm spectra, local charts, and selective lifting”
Explicit integrals now cover zeta derivatives, L-function values, and sphere determinants.
· “General Stirling-Ramanujan Constants are exponential periods and applications”
The same paper shows p-adic orthogonalization and closest-vector queries are unbounded in the worst case.
· “Norm-Query Complexity of Algorithmic Problems in Finite-Dimensional p-adic Normed Spaces”
Any non-empty exceptional set has positive 1/log(1/r)-Hausdorff measure; over finite fields the measure is infinite.
That makes them locally integrable over complex and mod-ell coefficients, uniformly.
The same window sum is 4C for almost all x unconditionally; the hypothesis makes it true for every large x.
A proof that every n/2-sized subset of {1,...,n} contains a wave of length c(log n)^2.
· “Dense ascending waves: A resolution of the Alon-Spencer conjecture”
Almost surely, the count matches the tube area times Q^3, except at one critical thickness.
If a non-abelian large sieve inequality holds, the count of primes with 2 as a primitive root is at least x/(log x)^2.
· “A non-abelian large sieve and Artin's primitive root conjecture”
The last open cases, where the characteristic divides the group order, are now settled.
· “The classification of generalised Kummer surfaces in positive characteristic”
New proof confirms Harper's conjecture, pinning the exact logarithmic exponent of the largest fluctuations.
· “Almost sure upper bound for sums of random multiplicative functions and critical chaos”
On odd prime powers, the new bound beats earlier exponentially weak savings unless shifts align.
· “Sums of products of Kloosterman sums to prime power moduli”
Dominant maps into such a family are classified by π1-open maps of profinite étale homotopy types over the base.