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Updated Oct 8

Problem 388

Statement. Can one classify all solutions of

\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)

where $k_1,k_2>3$ and $m_1+k_1\leq m_2$ ? Are there only finitely many solutions?

Status. Open.

Source. erdosproblems.com/388, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #388, https://www.erdosproblems.com/388.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

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Linked library material

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Linked from (11)

Diophantine Problems and Powers diophantine_problems/beukers_1999_irreducibility_polynomials_arithmetic_progressions_equal_products Erdős's conjecture as reported on p. 13: finitely many solutions of x(x+1)...(x+m-1) = λy(y+1)...(y+n-1)Theorem 1.1 (p. 13): equal products of two arithmetic progressions of fixed lengths and differences Theorem 2.1 (p. 15): when X(X+1)...(X+m-1) - λY(Y+1)...(Y+n-1) is reducible Theorem 2.2 (pp. 15--16): genus of the curve X(X+1)...(X+m-1) = λY(Y+1)...(Y+n-1)diophantine_problems/erdos_1976_problems_results_number_theoretic_properties_consecutive Theorem 3: n! = (m+1)...(m+t) has no solution with m < (2−ε)^n once n > n_0(ε)diophantine_problems/kulkarni_2005_class_diophantine_equations_involving_bernoulli_polynomials Theorem 1 (p. 52): a Bernoulli polynomial against a rising product Theorem C (p. 61): when x(x+1)...(x+m-1) = g(y) can have infinitely many solutions

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