Motivation

A unit fraction is the reciprocal $1/n$ of a positive integer. The number $1$ can be written as a sum of distinct unit fractions in infinitely many ways — $1 = \tfrac12+\tfrac13+\tfrac16$, $1 = \tfrac12+\tfrac14+\tfrac16+\tfrac1{12}$, and so on — and the combinatorics of such representations is one of the oldest recurring themes in Erdős's problem lists. Most questions in the area concern size: how many terms are needed, how small the largest denominator can be, how large the smallest one must be. Erdős Problem 287 asks instead about the shape of a representation: how tightly can the denominators be packed?

Order the denominators increasingly and look at their consecutive differences. For $1 = \tfrac12+\tfrac13+\tfrac16$ the differences are $1$ and $3$. The question is whether a difference of at least $3$ must always occur, in every representation of $1$, no matter how many terms it has. The problem is recorded in Erdős and Graham's 1980 problem book (ErGr80, p. 33) and was selected for the booklet of favourite problems prepared for the 1999 Budapest conference on Erdős's mathematics ([Va99, 1.15]). It remains open.

Timeline. The weaker statement that some difference must be at least $2$ — equivalently, that $1$ is never the sum of the reciprocals of a block of consecutive integers — is classical. Theisinger (1915) proved that the harmonic number $H_n$ is not an integer for $n \ge 2$, using Bertrand's postulate. Kürschák (1918) introduced the $2$-adic argument that proves the general block statement: for $m \le n-2$, the difference $H_n - H_m$ is not an integer. Erdős's 1932 paper [Er32], whose title translates as "A generalisation of an elementary number-theoretic theorem of Kürschák", extends the result from blocks of consecutive integers to arithmetic progressions; the erdosproblems.com entry for Problem 287 cites it for the difference-$\ge 2$ bound. Nothing stronger appears to be known: the passage from $2$ to $3$ is the open part, and no partial result is recorded in the entry beyond a conditional one, namely that the conjecture would follow for all but finitely many exceptions if it were known that for every large $N$ there is a prime $p \in [N, 2N]$ with $(p+1)/2$ also prime.

Setting

Fix an integer $k \ge 2$ and integers

$$1 < n_1 < n_2 < \cdots < n_k$$

with

$$1 \;=\; \frac{1}{n_1} + \frac{1}{n_2} + \cdots + \frac{1}{n_k},$$

the sum taken in $\mathbb{Q}$. Call such a tuple a representation of length $k$. The denominators are strictly increasing, hence distinct, and all exceed $1$: the value $n_1 = 1$ is excluded because $1/1$ already exhausts the total. The gaps of the representation are the $k-1$ consecutive differences $n_{i+1} - n_i$ for $1 \le i \le k-1$, and its maximal gap is $\max_i (n_{i+1} - n_i)$.

Representations exist for every $k \ge 3$, and for $k = 1$ only the excluded $n_1 = 1$; no representation of length $2$ exists. Examples: $(2,3,6)$ with gaps $1, 3$; $(2,4,6,12)$ with gaps $2,2,6$; $(3,4,6,10,12,15)$ with gaps $1,2,4,2,3$.

Formalization targets

Goal — Erdős Problem 287

$$\text{every representation } 1 < n_1 < \cdots < n_k \ (k \ge 2) \text{ of } 1 \text{ satisfies } \max_{1 \le i < k} (n_{i+1} - n_i) \;\ge\; 3.$$

This is the open conjecture, stated with no bound on $k$ and no restriction on the denominators beyond those in Setting. It is the weakest form that captures the question: asserting a bound for one particular $k$, or for denominators in some range, would be a different and strictly easier statement.

Milestone — the gap-two bound (Kürschák; Erdős [Er32])

$$\text{every representation satisfies } \max_{1 \le i < k}(n_{i+1} - n_i) \;\ge\; 2.$$

Equivalently: no block of two or more consecutive integers has reciprocals summing to $1$. This is closed mathematics and the natural first target.

Milestone — the classical block theorem (Kürschák)

$$\text{for } n \ge 1 \text{ and } k \ge 2, \qquad \sum_{i=0}^{k-1} \frac{1}{n+i} \notin \mathbb{Z}.$$

The gap-two bound is an immediate consequence, since a representation all of whose gaps equal $1$ is exactly a block of consecutive integers.

Milestone — sharpness

$$1 = \tfrac12+\tfrac13+\tfrac16 \text{ is a representation all of whose gaps are at most } 3.$$

So the constant $3$ in the goal is optimal and cannot be replaced by $4$.

Significance

The result itself. A positive answer would say that a representation of $1$ by unit fractions can never have all its denominators within distance $2$ of each other — a structural constraint of a kind that the size-based results in this area do not provide. The conditional route recorded on the problem page is instructive about where the difficulty sits: it reduces the conjecture, up to finitely many exceptions, to the existence of primes $p$ in $[N,2N]$ with $(p+1)/2$ prime, a statement of Bertrand-with-extra-structure type that is itself out of reach of current technology. A direct proof would therefore either bypass that route or resolve the conjecture for the remaining cases by different means.

Formalizing it. The gap-two bound and the block theorem behind it are closed mathematics, so the honest description of that part of this mission is formalization, not research. It is nevertheless not already available: Mathlib proves Theisinger's case harmonic_not_int, that $H_n \notin \mathbb{Z}$ for $n \ge 2$, but not Kürschák's block version $H_n - H_m \notin \mathbb{Z}$, which is the form Problem 287 needs. Supplying it is a genuine strengthening of the library's existing development and is reusable for any question about reciprocal sums over intervals. The goal itself is open, and this mission does not claim otherwise: it is registered with an open proof, and the milestones are what a solver can realistically close today.

Difficulty

The obvious first idea — bound the number of terms, then check finitely many cases — fails immediately, because $k$ is unbounded: representations of $1$ exist with arbitrarily many terms, so no finite computation can settle the conjecture. The second idea, extending the $2$-adic argument that gives the gap-two bound, also fails, and instructively. That argument works because a block of consecutive integers contains exactly one element of maximal $2$-adic valuation, which leaves the total with negative valuation. Once gaps of size $2$ are permitted the denominators may be chosen to avoid that configuration — for instance all even, as in $(2,4,6,12)$ — and the valuation obstruction disappears. There is no evident replacement prime or weighting that rules out all gap-$\le 2$ configurations simultaneously, and the conditional result quoted above suggests why: the known routes pass through the distribution of primes in short intervals with a multiplicative side condition, rather than through a congruence obstruction.

Formalization scope

A representation is encoded as a function $f : \mathbb{N} \to \mathbb{N}$ together with the hypotheses ∀ i < k, 1 < f i and ∀ i j, i < j → j < k → f i < f j, and the requirement ∑ i ∈ Finset.range k, (1 : ℚ) / f i = 1. Only the values of $f$ below $k$ are constrained; the function is not required to be monotone or bounded elsewhere, and nothing outside the window is used. The conclusion is ∃ i, i + 1 < k ∧ 3 ≤ f (i + 1) - f i, the existential form of "the maximal gap is at least $3$"; the subtraction is natural-number subtraction, which is harmless because $f$ is increasing on the window, so no truncation can occur. The sum is a rational equality, not an approximation.

The statement admits no trivializing reading. The hypothesis 1 < f i is essential and is not vacuous — dropping it would admit $f\,0 = 1$, $k = 1$; the strict monotonicity is what makes the gaps well defined and the denominators distinct; and k ≥ 2 guarantees that at least one gap exists, so the conclusion is not an empty existential. Asserting exactly $3$ rather than at least $3$ would be false, as $(2,4,6,12)$ has a gap of $6$.

Infrastructure: the block theorem is proved from Mathlib's padicNorm and padicValNat API — padicNorm.add_eq_max_of_ne, padicNorm.sum_lt', padicNorm.not_int_of_not_padic_int, pow_padicValNat_dvd and pow_succ_padicValNat_not_dvd — and needs no new definitions. That development is reusable beyond this mission and is a candidate for upstreaming to Mathlib alongside harmonic_not_int. Contributions are welcome on any milestone independently; a formalization of the conditional reduction to primes $p$ with $(p+1)/2$ prime would also be a valuable addition, and is not included as a milestone here only because the problem page states it too briefly to formalize faithfully without consulting a primary source.

Selected references

• P. Erdős, Egy Kürschák-féle elemi számelméleti tétel általánosítása (A generalisation of an elementary number-theoretic theorem of Kürschák), Mat. és Phys. Lapok 39 (1932), 17–24.
• P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique, Geneva, 1980, p. 33. scan
• Various, Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999, item 1.15.
• K. Conrad, The $p$-adic growth of harmonic sums, expository notes (Theorem 2 is Kürschák's block theorem, with the $2$-adic proof). pdf
• T. F. Bloom, Erdős Problem #287, erdosproblems.com/287.