Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Number theory
L103_totient_tail_cofactor_factorization/: Conditional on the unrefereed totients manuscript at two cuts, the tail totients of its coefficient factor through one prime tuple and a coprime cofactor outside vanishing mass, so the Problem 417 quotient is a limiting average of the cofactor's reciprocal least-preimage ratio.
L104_shifted_prime_totient_fibers/: The exceptional-prime theorem of Banks, Ford, Luca, Pappalardi and Shparlinski implies that all but x/(log x)^(2-delta) primes up to x have basin {p,2p} under phi(n)+1. Their union has integer density zero.
L105_totient_inverse_chain_cofactors/: An infinite increasing odd inverse chain for phi(n)+1 cannot have an eventually periodic largest-prime cofactor sequence, or have that cofactor eventually at most 9. A repeated block has an explicit 2-adic length bound; nonperiodic sequences remain uncontrolled.
L154_local_lucas_valuation/: When the first quotient digit is at least a polynomial's local multiplicity, full Lucas domination at one prime still permits a column with exactly the initial valuation gain.
L198_three_layer_prime_blocks/: Primes indexed by integer cells and placed by Chinese remaindering give a block of floor(alpha P) integers, 3<=alpha<4, whose multiples are exactly three layers counted by three projections of the cells.
L199_separated_grid_third_layer/: Extending a rectangular prime-grid block with separated rows past three times its largest prime adds at least half as many new locations as primes, so this construction costs at least linearly.
L200_long_block_sums_differences_exponent/: For 3<=alpha<4 the least number of distinct prime multiples in a block of floor(alpha P) integers grows like u^(1/beta+o(1)), where beta is the best ratio log|S|/log of the largest of three projections of an integer cell set.
L201_type_class_long_block_bound/: Three-cell type classes give an explicit exponent theta=log(27/4)/log 27 for long prime-multiple blocks, with a small logarithmic factor under Green--Tao realization and an elementary alternative using the prime number theorem.
L202_prime_realization_three_projections/: Every finite integer cell pattern with distinct differences has a dilated prime realization with exactly its three projection counts, giving the arithmetic Kakeya comparison for all length ratios from three up to four.
L203_rational_entropy_prime_block_bound/: A rational measure on Lemm's seven-point sums–differences configuration gives an absolute bound C u^0.56213 for the minimum distinct-multiple count, uniformly over all length ratios from three up to four.
L204_cartesian_prime_grid_obstruction/: Unique row-plus-column sums force a large third projection; even sliding a long block while retaining its row and column locations costs at least a constant times the number of grid primes to the two-thirds power.
L212_bounded_ratio_dissociated_subset/: A positive real set of ratio at most 2 has fewer than 3 times 2^d elements, and ratio at most 4/3 gives the sharp positive bound |P| < 2^d. A quantitative bound also uses the least possible nonzero coefficient sum of a relation.
L213_triple_log_dissociated_subset/: Every set of n >= 5 reals has a dissociated subset of size at least log_2(n-1) - log_2(ceil(log_2 log_2 n) + 2 + sqrt 2) - log_2 6, so f(n) >= log_2 n - log_2 log_2 log_2 n - O(1).
L22_near_logarithmic_dissociated_subset/: Every set of n real numbers contains an additively dissociated subset of size at least log_2 n - log_2(log_2 n + 1); the binary residue-tree bound of the Problem 963 research, weaker than the asked floor(log_2 n).
L23_central_binomial_product_density/: For each fixed k, squared upward products dividing the central binomial coefficient and downward products at the shifted argument differ on a set of natural density zero.
L24_powerful_sum_bound/: A quadratic norm count and dyadic interpolation give at most N^(2/5+o(1)) ordered representations of N as a sum of two positive powerful integers.
L270_recentered_pivot_obstruction/: For every R > 1, a positive odd set of ratio below R and size 2^(k-1)+1 can have no recentered graded certificate with k companions. A 65-element example nevertheless contains eight dissociated elements.
L34_exceptional_widths_for_omega/: A second-moment estimate permits widths to be enlarged on a density-zero set while keeping every normalized local count bounded.
L35_divisor_prefix_compression/: An exact finite partition formula computes the least integer realizing a prescribed divisor-prefix sum, with complete values through 64.
L37_prime_multiple_block_extremum/: A two-half lower bound and a rectangular prime-grid construction give the exact minimum over prime sets for every number of primes.
L43_continuous_totient_phase_ratio/: Under the unrefereed totients manuscript and L65, the phase coefficients of V and V' are continuous with the same jump factor at the period boundary, so the Problem 417 quotient is a continuous function on the circle.
L44_finite_prime_totient_phase/: Conditional on the unrefereed totients manuscript, least-preimage divisibility counts have continuous phase profiles, and finite-prime weighted sums approximate the Problem 417 quotient uniformly in phase.
L53_divisor_product_collision_criterion/: Collision decay exponent above one half for products of odd divisors of K! modulo powers of two would give Problem 18(c) with no exponential Fourier loss; nonconcentration at every level cannot supply that rate.
L63_polylogarithmic_dyadic_representatives/: A conditional criterion for Problem 18(c): dyadic residue sums with a factorial index linear up to logarithms, polylogarithmic length, and a polylogarithmic height exponent imply a polylogarithmic bound for h(n!).
L64_factorial_digit_entropy/: A counting obstruction to fixed positive dyadic nonconcentration for polylogarithmically many separated divisor digits with factorial index linear up to logarithms in the total bit length.
L65_conditional_totient_ratio_coefficient/: Assuming the unrefereed totients manuscript's structural estimates, the weight 1/r gives the coefficient for distinct totients with a preimage at most x, reducing Problem 417 exactly to constancy in the phase.
L66_divisor_residue_spread/: Pairing divisor exponents gives a dyadic concentration bound for all odd divisors of K!; polynomial quantitative bounds in the weighted mixing theorem would then imply the polylogarithmic factorial bound.
Native claims filed under the number-theory problems of erdosproblems.com. Each card carries its exact statement, its standing and tier, and its formal surface.