Erdős research

Computational studies in graph theory and number theory. Explore the topics and review status of ongoing work.

No entry claims a solution to an unrestricted problem. Illustrations show general mathematical ideas, not unpublished results.

A triangle, square, and pentagon illustrate different cycle lengths.

Problem 84 / Graph theory

Cycle lengths in finite graphs

Studies which combinations of cycle lengths can occur in finite graphs. A manuscript is awaiting expert review.

A sequence of consecutive integers leads to a group of prime factors.

Problem 382 / Number theory

Prime factors of consecutive products

Studies prime factors in products of consecutive integers. Candidate findings await specialist review and comparison with published work.

A highlighted finite segment within several continuing sequences of dots.

Problem 699 / Number theory

Testing a conjecture over finite ranges

Uses finite computation to investigate a number-theory conjecture. Independent verification remains pending; a finite check cannot prove the unrestricted statement.

Multiples of three and five highlighted along a number line.

Problem 488 / Additive number theory

The distribution of multiples

Studies how multiples of selected integers are distributed across intervals. Draft mathematical work remains unreviewed and does not settle the general problem.

Radial groups and a dot lattice illustrate divisors and frequency.

Problem 859 / Number theory

Divisor sums and density

Studies how often conditions involving divisor sums occur. Draft findings await expert review and comparison with published work.

Selected dots within overlapping circles illustrate set membership.

Problem 1074 / Number theory

Membership in a number set

Studies whether individual numbers belong to the set in this problem. Partial findings await specialist review and do not establish its overall density.

Different arrangements of square tiles illustrate integer partitions.

Problem 1106 / Number theory

Prime factors and integer partitions

Studies prime factors associated with integer partitions. Finite computations await expert review and do not settle the unrestricted problem.

Two branching structures end in matching groups of colored nodes.

Problem 850 / Number theory

Shared prime-factor structure

Investigates when distinct integers share a prime-factor structure. Exploratory computations await independent verification and do not resolve the problem.

Two offset groups of tiles illustrate blocks of consecutive integers.

Problem 686 / Number theory

Products of consecutive integers

Investigates relationships between products of consecutive integers. Exploratory findings require further review and do not resolve the general problem.

Three small networks illustrate different graph construction patterns.

Problem 617 / Graph theory

Limits of graph constructions

Studies the limits of selected graph constructions. Draft findings concern a restricted family and require specialist review; they do not resolve the general problem.

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