Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

Proof manifests naming this Problem

• William Blair Lean proofswilliamjblair:Erdos730.audit_unitBand_complement_iff
• William Blair Lean proofswilliamjblair:Erdos730.audit_unitBand_slack_ge_iff
• William Blair Lean proofswilliamjblair:Erdos730.audit_unitBandEnvelope_iff
• William Blair Lean proofswilliamjblair:Erdos730.ConsecutiveTransition.consecutive_primeFactors_eq_iff_transitionConditions
• William Blair Lean proofswilliamjblair:Erdos730.ConsecutiveTransition.exists_obstruction_of_primeFactors_ne
• William Blair Lean proofswilliamjblair:Erdos730.cutoff_lt_of_unitBand_maximal
• William Blair Lean proofswilliamjblair:Erdos730.densityBudget_final_lt
• William Blair Lean proofswilliamjblair:Erdos730.dyadicThresholdBase_strictMono_step
• William Blair Lean proofswilliamjblair:Erdos730.eight_mul_sq_le_succ_cube
• William Blair Lean proofswilliamjblair:Erdos730.endpoint_payment_identity
• William Blair Lean proofswilliamjblair:Erdos730.endpoint_payment_lt_one_percent
• William Blair Lean proofswilliamjblair:Erdos730.endpoint_powered_threshold_certificate
• William Blair Lean proofswilliamjblair:Erdos730.finite_geometric_prime_power_tail
• William Blair Lean proofswilliamjblair:Erdos730.finite_prime_power_pair_count
• William Blair Lean proofswilliamjblair:Erdos730.finite_reciprocal_square_tail
• William Blair Lean proofswilliamjblair:Erdos730.finite_reciprocal_tail_from_root_envelopes
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.first_power_fixed_upper_slope
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.improved_higher_power_ceiling_lt_three_tenths
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.normalized_block_cover_six_fifths
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.q_top_low_digit_large
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.q_top_small_cofactor_of_square_lt
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.q_top_two_digit_large
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.remaining_first_power_budget
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.s_top_low_digit_large
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.s_top_small_cofactor_of_square_lt
• William Blair Lean proofswilliamjblair:Erdos730.FirstPowerRoutes.s_top_two_digit_large
• William Blair Lean proofswilliamjblair:Erdos730.FullDensityCore.branches_pairwise_coprime
• William Blair Lean proofswilliamjblair:Erdos730.FullDensityCore.product_identity
• William Blair Lean proofswilliamjblair:Erdos730.FullDensityCore.switching_allowed_class_count_certificate
• William Blair Lean proofswilliamjblair:Erdos730.FullDensityTheorem.pairSet_infinite
• William Blair Lean proofswilliamjblair:Erdos730.halfBand_endpoint_payment_lt_one_percent
• William Blair Lean proofswilliamjblair:Erdos730.halfBand_endpoint_powered_threshold_certificate
• William Blair Lean proofswilliamjblair:Erdos730.halfBandEnvelope_forces_high_exponent
• William Blair Lean proofswilliamjblair:Erdos730.halfBandEnvelope_prime_power_clearance
• William Blair Lean proofswilliamjblair:Erdos730.halfDigitParity_probabilityError
• William Blair Lean proofswilliamjblair:Erdos730.higherPrimePowerPairs_card_le
• William Blair Lean proofswilliamjblair:Erdos730.KummerTransition.not_dvd_centralBinom_iff_lowerHalfDigits
• William Blair Lean proofswilliamjblair:Erdos730.nearEnvelope_forces_high_exponent
• William Blair Lean proofswilliamjblair:Erdos730.nearEnvelope_prime_power_clearance
• William Blair Lean proofswilliamjblair:Erdos730.ObstructionMaps.PhiP_root_progression
• William Blair Lean proofswilliamjblair:Erdos730.ObstructionMaps.PhiQ_root_progression
• William Blair Lean proofswilliamjblair:Erdos730.ObstructionMaps.PhiR_root_progression
• William Blair Lean proofswilliamjblair:Erdos730.ObstructionMaps.PhiS_root_progression
• William Blair Lean proofswilliamjblair:Erdos730.ObstructionMaps.prime_dvd_residual_support
• William Blair Lean proofswilliamjblair:Erdos730.padicBranchAllowedCount_le
• William Blair Lean proofswilliamjblair:Erdos730.padicBranchMap_bijective
• William Blair Lean proofswilliamjblair:Erdos730.powered_threshold_of_halfBand_maximal
• William Blair Lean proofswilliamjblair:Erdos730.powered_threshold_of_near_maximal
• William Blair Lean proofswilliamjblair:Erdos730.restrictedDigitBox_card
• William Blair Lean proofswilliamjblair:Erdos730.tendsto_higherPrimePowerPairs_card_div
• William Blair Lean proofswilliamjblair:Erdos730.tendsto_tsum_higherPower_of_dominated
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_cuberoot_floor_certificate
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_payment_identity
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_payment_lt_one_percent
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_payment_margin
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_sqrt_floor_certificate
• William Blair Lean proofswilliamjblair:Erdos730.unitBand_endpoint_threshold_certificate
• William Blair Lean proofswilliamjblair:Erdos730.unitBandDyadicThresholdBase_strictMono_step
• William Blair Lean proofswilliamjblair:Erdos730.unitBandEnvelope_forces_high_exponent
• William Blair Lean proofswilliamjblair:Erdos730.unitBandEnvelope_prime_power_clearance
• William Blair Lean proofswilliamjblair:Erdos730.UnitRangeBlock.higher_prime_power_payment_ceiling_lt_half
• William Blair Lean proofswilliamjblair:Erdos730.UnitRangeBlock.normalized_block_cover_cross_bound
• William Blair Lean proofswilliamjblair:Erdos730.UnitRangeBlock.quadratic_block_difference_dvd_sq
• William Blair Lean proofswilliamjblair:Erdos730.UnitRangeBlock.quadratic_block_expansion