Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Integer sequences
L123_odd_root_arc_colorings/: Coloring the N-th roots of unity by q arcs of argument gives dissociated classes exactly when q times the least degree of a {-1,0,1} multiple of the N-th cyclotomic polynomial is at least N; the normalized degree falls along divisibility, so a positive infimum bounds every odd prime grid.
L124_flat_cyclotomic_degree_ratio/: Open: the least degree of a {-1,0,1} multiple of the M-th cyclotomic polynomial is at least a fixed positive fraction of M for all odd squarefree M; a proof would bound the dissociated coloring number of every distinct-prime tensor grid of Problem 774.
L20_roots_of_unity_analytic_sidon/: One constant bounds every trigonometric polynomial supported on roots of unity by the value of some additive character of the discrete additive group of the complex numbers; a step of the Problem 774 extraction route.
L210_odd_root_three_colorings/: Open: for every odd squarefree N the N-th roots of unity split into three dissociated sets; a proof would close the distinct-prime tensor route to a counterexample for Problem 774 and answer Déchamps's covering question, and a single order needing four classes would refute it.
L211_fractional_half_cover_obstruction/: A thirteen-element positive integer set has hereditary dissociation ratio 7/13, fractional covering number 37/19, and dissociated covering number three; exact finite certificates separate weighted half-extraction from two-colorability.
L21_roots_of_unity_proportional_dissociation/: One integer denominator works for every finite set of complex roots of unity: each contains an additively dissociated subset of proportional size; the roots-of-unity analog of Problem 774's hypothesis.
L250_explicit_near_square_smooth_blocks/: For K >= 64 and 0 <= m <= K^(5/2)/128 the block m+1, ..., m+K contains c(q-t)(q+t) with all three factors in [1, K], so k(n) in Problem 962 is below max(64, (128n)^(2/5)); an explicit form of Erdős and Sárközy's 1979 exponent 2/5.
L251_averaged_near_square_smooth_blocks/: For K >= 64 and 0 <= m <= 2^(-15) K^(28/11) the block m+1, ..., m+K contains c(q-t)(q+t) with all three factors in [1, K]; so k(n) in Problem 962 is below max(64, (2^15 n)^(11/28)), improving the exponent 2/5 of Erdős and Sárközy to 11/28.
L252_nonresidue_blocks_without_smooth_members/: For a prime p = 3 (mod 4) with least quadratic nonresidue q, each of p-q+1, ..., p-1 has a prime factor at least q, so k(p) >= q-1 in Problem 962; an exponent below 1/(4 sqrt(e)) for k(n) would improve Burgess's bound for these primes.
L253_exponent_pair_smooth_blocks/: Bourgain's exponent pair applied to the averaged near-square construction gives a K-smooth integer in every block (m,m+K] with m <= K^(97/38-delta), for large K depending on delta, and hence k(n) <<_epsilon n^(38/97+epsilon) in Problem 962.
L269_simplex_affine_fiber_threshold/: In the k-fold tensor of a q-point simplex over a prime-power index field, every affine fiber with nonzero coefficients is rationally independent for k=q.
L271_repeated_simplex_extraction/: The integer relation lattice of a repeated power-of-two simplex is semistable, and its normalized determinant stays above one at a critical tensor scale, giving one hereditary dissociation constant through the stated stable-lattice and phase-entropy theorems.
Native claims filed under the integer-sequence problems of erdosproblems.com. Each card carries its exact statement, its standing and tier, and its formal surface.