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A D(n) pair of triangular numbers is a pair (Tₐ,T_b) with TₐT_b+n a perfect square, where Tₐ=a(a+1)/2. Bagchi and Zhou-Zheng prove that no such pair exists when n ≡ 2,5 (mod 9), observe computationally that n=12,17,42 also admit no pair with a ≤ 10⁵, conjec…

Let α be an algebraic number of degree d over ℚ. Dubickas asked, as Problem 11123 of the American Mathematical Monthly, whether three distinct conjugates of α can sum to zero when 3 ∤ d, and Stong's polynomial t²⁰+4 · 5⁹t¹⁰+16 · 5¹⁵ shows that they can. Bar…

Let p>3 be a prime, n=(p-1)/2, and let Cₚ(x)=[x+cᵢⱼ]_(1 ≤ i,j ≤ n) be the shifted Legendre-symbol matrix with c₁ⱼ=leg(j, p) and cᵢⱼ=leg(i-j, p) for i ≥ 2. Ren and Sun, having evaluated the companion matrix with leg(i+j, p) in place of leg(i-j, p), record a…

Let p be an odd prime, n=(p-1)/2, and let A(p)=[leg(j+k, p)+legj²+k²p]_(0 ≤ j,k ≤ n) be the integer matrix of Legendre symbols studied by Jiang and Sun. Sun's Conjecture 4.10(ii) asserts that 2det A(p) is a quadratic residue modulo p whenever p ≡ 3 (mod 4).…

For positive integers k and ℓ let R_(k,ℓ)(x)=Σⱼ₌₀ᵏ⁺¹(-1)^((ℓ+1)j)(tfracB₂ⱼ(2j)! tfracB₂ₖ₊₂₋₂ⱼ(2k+2-2j)!)^(ℓ)xʲ, the two-parameter family of reciprocal polynomials introduced by Maji and Sarkar [ms] and studied by Charan, Meher and Pathak [cmp], who prove th…

Bremner and Ulas reduce the divisibility of the quadrinomial xⁿ+xⁿ⁻ᵐ+xⁿ⁻²ᵐ+a by a rational quadratic to the rational points of the hyperelliptic curve s²=F_(m,n)(t), where F_(m,n)=Aₙ₋ₘ²-4Aₙ₋₂ₘAₙ is built from the sequence Aₖ=Aₖ(1,t) ∈ ℤ[t] of their Lemma 3.…

For an odd prime p let Qₚ be the set of nonzero quadratic residues modulo p and let 1={1}. A sequence (x₁,…,xₖ) in Zn(p)=ℤ/pℤ is a (Qₚ,1)-weighted zero-sum sequence if there are a₁,…,aₖ ∈ Qₚ with a₁x₁+…+aₖxₖ=0 and a₁+…+aₖ=0, and D_(Qₚ,1)(p) is the least k s…

For odd n put Sₙ(q)=Σₖ₌₀^((n-1)/2)((q;q²)ₖ²(q²;q⁴)ₖ)/((q²;q²)ₖ²(q⁴;q⁴)ₖ)(-q)ᵏ. Guo and Zudilin evaluated Sₙ modulo the cyclotomic polynomial Φₙ(q) for n ≡ 1 (mod 4) and left the two remaining residue classes as the only open problem of their paper: to show…

Let 𝔽_(q) be a finite field and let f=X²+bX+c ∈ 𝔽_(q)[X] be irreducible with c a primitive element of 𝔽_(q). Vega [vega2607] proved that if q+1=4π for an odd prime π, then f is a primitive polynomial if and only if b² ≠ 2c, and asked whether the hypothes…

Ross calls n an S-perfect number of the first kind when the divisors of n strictly between 1 and n can be given coefficients from S so that 1+Σⱼλⱼdⱼ=n; for S={-1,1} this is a signed-divisor, or subset-sum, condition. He conjectured that every odd nonsquare…

Vaughan and Wooley proved that every sufficiently large positive integer is a sum of at most 50 positive ninth powers, without an effective threshold. Benfield and Lippard, who computed the corresponding largest exception for the exponents k=5,6,7,8, left t…

For a positive integer N put ℓ_N=lcmop(1,2,…,N) and let S_N={P ∈ ℤ[x]:deg P<N, ∫₀¹P(x) dx=1/ℓ_N} be D. Bazzanella's set of integer polynomials with the smallest possible positive integral on [0,1]; it is the set on which the Gelfond–Schnirelman–Nair element…

Sawin, Shusterman and Stoll attach to a pair (c,d) of integer polynomials with c(0),d(0) ≠ 0 an integer m₀(c,d): the least M such that for every m>M, every pair (a,b) with deg a,deg b<m, ab ≡ cd mod xᵐ and wt(a)+wt(b) ≤ wt(c)+wt(d) already satisfies ab=cd,…

For a finite set T of positive integers let L_(T)=(lcm(tᵢ,tⱼ)) and G_(T)=(gcd(tᵢ,tⱼ)) be the LCM and GCD matrices on T. Merikoski, Haukkanen, Sasaki and Tossavainen conjecture that, on T={1,…,n} with n>3, the number -1 is a generalized eigenvalue of L_(T) t…

Let l be an odd prime, p ≡ 1 (mod l) a prime, γ a generator of 𝔽ₚ^(×), and let J(1,1)ₗ=Σᵢaᵢζₗⁱ be the Jacobi sum of order l attached to γ. Katre and Rajwade's condition (vi), the condition that pins down which conjugate of J(1,1)ₗ belongs to γ, becomes aft…

For relatively prime positive integers a,b exactly one of the two Diophantine equations ax+by=(a-1)(b-1)/2 and 1+ax+by=(a-1)(b-1)/2 has a nonnegative integral solution, and Γ(a,b) ∈ {0,1} records which one. Chu, Miller and Tresch (arXiv:2512.12681) write Tₖ…

Let Fₙ and Lₙ be the Fibonacci and Lucas polynomials, defined by F₀=0, F₁=1, L₀=2, L₁=x and the common recurrence Pₖ₊₂=xPₖ₊₁+Pₖ. A recent preprint of Chen, Guo and Hong proves that every irreducible factor of Fₙ is monogenic when n is odd and that every irr…

For n ≥ 1 let F_(R)(n) be the n × n integer matrix with (i,j) entry 1 if j=1, Fᵢ if i | j, and 0 otherwise, where Fᵢ is the i-th Fibonacci number: the Fibonacci-weighted analogue of Redheffer's matrix introduced by Doumas and Psarrakos. They prove that the…

Let Dₙ(x,α) be the Dickson polynomial of the first kind over 𝔽_q and let D_(n,α) be the self-map of 𝔽_q it induces. Under the hypothesis αⁿ=α, which makes D_(n,α)ᵐ = D_(nᵐ,α), Chen and Peng (arXiv:2508.08621v3) compare the exact period k of the integer se…

For a positive integer n let B(n) be the largest height of a polynomial in ℤ[x] dividing xⁿ-1, the height of a polynomial being its largest coefficient in absolute value. Pomerance and Ryan determined B(pᵏ) and B(pq), and Kaplan determined B(p²q); the first…

For a nonzero integer n, a D(n)-set is a set of distinct nonzero integers whose pairwise products, increased by n, are all perfect squares. Adžaga, Dujella, Kreso and Tadić asked, for each k, how small in absolute value a nonzero integer n₁(k) can be when s…

A Chebyshev-type quadrature of degree m, or an m-design, for a probability measure w(t) dt on an interval I is a set of distinct points of I over which the unweighted average of every polynomial of degree at most m equals its integral. For the Chebyshev mea…

Let w(z)=-e^(-2/z)/(4π i) be the weight on the unit circle for which the Bessel polynomials are orthogonal, and let K=ℚ(i) ∩ S¹ be the set of rational points of the circle. Matsumura asks, in part (2) of his Problem 1.2, whether for a given r there is a qua…

Let Eₙ be the number of alternating permutations of {1,…,n}, so that sec z+tan z=Σ_(n ≥ 0)Eₙzⁿ/n!. Knuth and Buckholtz proved that (Eₙ mod q) is eventually periodic; write d(q) for its minimal eventual period and s(q) for its preperiod. Ramassamy conjecture…