Erdős Problem #316

Erdős Problem #316: two bins of unit fractions

Complete reconstruction of Sándor's counterexample, 23 September 2026.

1) Statement

If a finite set $A\subset\mathbb N\setminus\{1\}$ satisfies $\sum_{d\in A}1/d<2$, must it be partitionable into two sets, each with reciprocal sum strictly less than one? The answer is no. The strict inequalities matter, and the counterexample uses distinct denominators.

2) The explicit denominator set and its sum

Take all divisors of 120 strictly between 1 and 120: \[ A=\{2,3,4,5,6,8,10,12,15,20,24,30,40,60\}. \] In particular neither denominator one nor denominator 120 is included. Multiplication by 120 gives the integer weights \[ \{60,40,30,24,20,15,12,10,8,6,5,4,3,2\}, \] whose sum is $239$. Equivalently, the divisor sum is \[ \sigma(120)=(1+2+4+8)(1+3)(1+5)=360, \] so the sum of reciprocal divisors after removing the terms for 1 and 120 is $360/120-1-1/120=239/120<2$.

3) The denominator lattice prevents a partition

Every subset sum of the reciprocals in $A$ is an integer multiple of $1/120$. If such a sum is strictly less than one, it is at most $119/120$. Therefore two parts each satisfying the required strict bound would have total at most \[ \frac{119}{120}+\frac{119}{120}=\frac{238}{120}, \] contradicting the actual total $239/120$. This argument includes empty parts and exhausts all possible partitions without a subset search.

More generally, if every denominator of a finite set divides an integer $L$, any partition into $r$ parts of reciprocal sum less than one forces total sum at most $r-r/L$. Thus any total strictly between $r-r/L$ and $r$ gives an obstruction. This is a criterion; no construction for every $r$ is inferred from the single example above.

4) Outcome and source

SOLVED by an explicit counterexample. The total, distinctness, exclusion of denominator one and impossibility of every two-part partition are proved exactly. This is Sándor's known example, not a new counterexample or a claim that fourteen denominators are minimal.

Source: https://www.erdosproblems.com/316, checked 23 September 2026, which attributes the divisor-of-120 construction to Sándor $1997$. The phrase "proper divisors" has been replaced above by the full list to remove any ambiguity about the excluded endpoints.

The estimate measures complete resolution of the stated two-part question, not a correctness probability.

5) COMPLETION ESTIMATE

COMPLETION: 100%