Optimal bounds for an Erdős problem on matching integers to distinct multiples

OPTIMAL BOUNDS F OR AN ERDŐS PR OBLEM ON MA TCHING INTEGERS TO DISTINCT MUL TIPLES W OUTER V AN DOORN, Y ANY ANG LI, AND QUANYU T ANG Abstra ct. Let f ( m ) b e the largest integer such that for every set A = { a 1 < · · · < a m } of m p ositive in tegers and every op en interv al I of length 2 a m , there exist at least f ( m ) disjoint pairs ( a, b ) with a ∈ A dividing b ∈ I . Solving a problem of Erdős, we determine f ( m ) exactly , and show f ( m ) = min  m, ⌈ 2 √ m ⌉  for all m . The proof w as obtained through an AI-assisted workflo w: the proof strategy was first prop osed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exp osition and final pro ofs presented here are en tirely h uman-written. 1. Intr oduction F or each p ositive integer m , let f ( m ) denote the largest integer r with the following prop erty: whenev er A = { a 1 < a 2 < · · · < a m } is a set of m positive in tegers, then for ev ery real n um b er x there exist distinct elemen ts c 1 , . . . , c r ∈ A and distinct in tegers b 1 , . . . , b r suc h that x < b i < x + 2 a m and c i | b i (1 ≤ i ≤ r ) . In their 1959 pap er [10], Erdős and Surán yi pro ved the first result on f ( m ) , obtaining the lo w er b ound f ( m ) ≥ √ m . In the opp osite direction, Erdős and Selfridge [7, 8] show ed that f ( m 2 ) ≤ 2 m holds for all m , from whic h one can deduce the general upp er b ound f ( m ) ≤ 2 ⌈ √ m ⌉ . Hence, known lo w er and upp er b ounds on f ( m ) differ b y a factor of t w o, and it is natural to wonder what the exact growth rate is. In [9], Erdős in particular asked whether the lo w er b ound can b e improv ed, and estimating f ( m ) is now recorded as Erdős Problem #650 on Blo om’s website [2]. In this pap er, w e determine f ( m ) exactly , thereby completely resolving the problem. T aking a step bac k, one migh t wonder why 2 a m w ould b e a natural choice for the length of the in terv al in the definition of f ( m ) . F or a start, if we instead let the length of the interv al b e ca m for some 1 < c < 2 , then the corresp onding definition of f ( m ) would actually give f ( m ) = 1 for all p ositiv e integers m . Indeed, such an interv al certainly contains an integer divisible by a 1 , giving the low er b ound f ( m ) ≥ 1 . F or the corresp onding upp er b ound one can c hoose a large in teger M , c ho ose A = { a 1 , a 2 , . . . , a m } with a i = M + i for all i , and set x = − a 1 + Q m i =1 a i . If M is sufficien tly large in terms of c and m , then one can v erify that, with 1 < c < 2 , the only in teger in ( x, x + ca m ) divisible by some element of A is x + a 1 . F or c ≥ 3 on the other hand, Erdős writes in [7] and [8] that “all hell breaks lo ose”. F or such v alues of c it is unclear what to exp ect exactly , although in [8] a quic k argument is recorded sho wing that an y interv al of length 3 a m con tains more than √ 6 m distinct multiples of the a i . Finally , 2020 Mathematics Subje ct Classific ation. Primary 11B75; Secondary 05C70, 05C35. Key wor ds and phr ases. com binatorial num ber theory , multiples in interv als, Hall’s marriage theorem, Erdős Problem. 1 2 W. V AN DOORN, Y. LI, AND Q. T ANG ev en though we defined f ( m ) using an interv al length of 2 a m , the results in this pap er will actually co v er the entire intermediate range 2 ≤ c < 3 . Indeed, in analogy with [7, 8], it will b ecome clear that our pro ofs still go through with 2 increased to an y real smaller than 3 . No w, to b etter understand f ( m ) , it is con v enient to reform ulate its definition in graph- theoretic terms. W riting N = { 1 , 2 , 3 , . . . } as the set of positive integers, let a set A ⊆ N (with | A | = m and max A = a m ) and a real num b er x b e given. With B = ( x, x + 2 a m ) ∩ Z , w e then define a bipartite graph G ( A, x ) with vertex classes A and B , joining a ∈ A to b ∈ B whenev er a | b . Let F ( A, x ) denote the size of a maxim um matching in G ( A, x ) . Then f ( m ) = min | A | = m min x ∈ R F ( A, x ) , where the outer minimum is ov er all m -elemen t sets A ⊆ N . This viewpoint will be esp ecially relev ant in the proof of the lo wer bound. 1.1. AI-assisted pro of disco v ery and formal v erification. A distinctive feature of the presen t paper is the workflo w b y whic h the proof w as obtained and v erified. An initial proof draft w as pro duced by ChatGPT (model: GPT-5.4 Pro), 1 whic h correctly identified the main pro of strategy but left a gap in one of the detailed arguments. A public record of this stage of the process is av ailable in the discussion thread for Erdős Problem #650; see [3]. The draft argumen t w as then given to Aristotle, 2 whic h supplied the missing details and pro duced a complete Lean formalization of the proof. F or transparency , an early proof draft and a subsequen t note iden tifying a gap in that draft are publicly a v ailable; see [15, 16]. More information on the formalization pro cess can be found in Section 5. This example suggests a possible researc h workflo w in which a large language mo del is used primarily for pro of search and high-lev el pro of discov ery , while a formal-reasoning system is used to turn the resulting argumen t into a fully rigorous and mac hine-chec ked pro of. Suc h a w orkflo w do es not remo ve human judgmen t altogether (for example, the original pro ofs w ere o v erly lengthy and ha v e, through h uman rewriting, decreased in size b y ab out a factor of t w o), but it can substantially reduce the extent to which correctness dep ends on traditional line-b y-line manual pro of c hecking. 1.2. P ap er organization. In Section 2 w e state the main result and explain ho w the exact form ula follows from separate upp er and lo w er b ounds. Section 3 pro v es the upp er b ound b y constructing, for an y pro duct m = st , a set A of m integers and an interv al of length 2 max A in which at most s + t distinct multiples o ccur. Section 4 establishes the matc hing low er b ound using a generalization of Hall’s theorem, together with a careful analysis of the neighbourho o d sizes in the relev an t bipartite graph. Finally , Section 5 briefly records the process b y which we obtained a formal verification of the argumen t in Lean. 2. St a tement and deduction of our main resul t The main result we pro v e is the following. 3 ✓ ✓ Theorem 2.1. F or every p ositive inte ger m we have f ( m ) = min  m, ⌈ 2 √ m ⌉  . 1 GPT is a large language mo del developed by Op enAI; for background on the GPT-5 family , see [14]. 2 Aristotle is an AI system for formal mathematical reasoning developed by Harmonic; see [1]. F or the pro ject page, see [11]. 3 Throughout, the sym b ol ✓ ✓ indicates that the pro of of the corresponding statement has b een formalized in Lean 4. MA TCHING INTEGERS TO DISTINCT MUL TIPLES 3 ✓ ✓ R emark 2.2 . As ⌈ 2 √ m ⌉ ≤ m for all m ≥ 4 , Theorem 2.1 in particular implies that f ( m ) = ⌈ 2 √ m ⌉ for all m ≥ 4 . No w, from the definition of f ( m ) , the upp er b ound f ( m ) ≤ m is immediate. A ccordingly , the pro of of Theorem 2.1 naturally splits into sho wing the upp er b ound f ( m ) ≤ ⌈ 2 √ m ⌉ for all m on the one hand, and the lo wer bound f ( m ) ≥ min( m, ⌈ 2 √ m ⌉ ) on the other. Giv en positive in tegers s and t , for the upper b ound we construct a sp ecific set A with | A | = st and an open in terv al I of length 2 max A , such that I contains at most s + t distinct m ultiples of elemen ts of A . In particular, any matching in the corresponding bipartite graph has size at most s + t , and hence f ( st ) ≤ s + t. (1) T o deduce from this that f ( m ) ≤ ⌈ 2 √ m ⌉ for ev ery m , let k b e suc h that k 2 < m ≤ ( k + 1) 2 . If k 2 < m ≤  k + 1 2  2 , then m ≤ k ( k + 1) and ⌈ 2 √ m ⌉ = 2 k + 1 . Since f ( m ) is monotone non-decreasing in m (indeed, one ma y alwa ys pass to a subset), we ha v e f ( m ) ≤ f ( k ( k + 1)) . Using the b ound (1) with s = k and t = k + 1 , we obtain f ( m ) ≤ f ( k ( k + 1)) ≤ 2 k + 1 = ⌈ 2 √ m ⌉ . If instead  k + 1 2  2 < m ≤ ( k + 1) 2 , then ⌈ 2 √ m ⌉ = 2 k + 2 . In this case w e tak e s = t = k + 1 in (1), and similarly deduce f ( m ) ≤ f (( k + 1) 2 ) ≤ 2 k + 2 = ⌈ 2 √ m ⌉ . W e remark that (1) is ever so sligh tly stronger than the earlier bound f ( m 2 ) ≤ 2 m prov ed b y Erdős and Selfridge [7, 8]. Indeed, their bound yields f ( m ) ≤ 2 ⌈ √ m ⌉ , whic h in general is w eak er than f ( m ) ≤ ⌈ 2 √ m ⌉ . Ev en though these estimates differ b y at most 1 , this difference do es of course b ecome relev an t if one wan ts to determine f ( m ) precisely . As for the lo wer b ound, we create the bipartite graph that w e introduced in Section 1. W e then recall that, for a subset S ⊆ A , the neigh bourho o d Γ( S ) ⊆ B of S is defined as the set of vertices in B that are joined by an edge to an element in S . W e then ha ve the following generalization of Hall’s theorem (see [4, Exercise 16.2.8(b)]): ✓ ✓ Lemma 2.3. F or any finite bip artite gr aph with vertex sets A and B , ther e exists a matching of size min  | A | , | A | − max ∅ = S ⊆ A ( | S | − | Γ( S ) | )  , wher e the inner maximum is taken over al l non-empty subsets S ⊆ A . R emark. As in [4], the ab ov e expression is also sometimes referred to as the König–Ore formula, and is generally more simply written as | A | − max S ⊆ A ( | S | − | Γ( S ) | ) instead, where the maxim um is tak en o v er al l subsets S ⊆ A . This is quickly seen to b e equiv alent to our v ersion, b y considering the empty set separately . With Lemma 2.3, we claim that the desired inequalit y then follows from sho wing that for all S ⊆ A w e hav e | Γ( S ) | ≥ 2 p | S | . (2) T o see wh y this w ould be sufficien t, note that t − 2 √ t is an increasing function on N , due to the fact that the inequalit y 2 √ t + 1 − 2 √ t < 1 holds for all t ≥ 1 . With | A | = m , inequalit y (2) therefore implies | S | − | Γ( S ) | ≤ | S | − 2 p | S | ≤ m − 2 √ m 4 W. V AN DOORN, Y. LI, AND Q. T ANG for all non-empt y S ⊆ A , so that by Lemma 2.3 we deduce the existence of a matching of size at least min( m, 2 √ m ) . As f ( m ) is an in teger, this indeed implies f ( m ) ≥ min( m, ⌈ 2 √ m ⌉ ) . W e will accomplish the proof of (2) by first splitting any in terv al of length 2 max A into t wo halv es. W e then define an injection from S in to the pro duct of the tw o separate halves of Γ( S ) , in order to obtain the required lo wer bound on | Γ( S ) | . 3. The upper bound As explained in Section 2, for the upp er b ound it is sufficient to sho w the following tw o- parameter inequalit y . ✓ ✓ Theorem 3.1. W e have f ( st ) ≤ s + t for al l p ositive inte gers s and t . Pr o of. If min( s, t ) = 1 , then the upp er bound f ( st ) ≤ st < s + t trivially holds. W e will therefore assume s, t ≥ 2 , and by symmetry w e ma y further assume s ≤ t . As men tioned in Section 2, w e now aim to construct a set A with st elements for which an in terv al of length 2 max A exists that contains at most s + t multiples of the elemen ts in A . In order to construct A , we let D b e the least common m ultiple of the integers 1 , 2 , . . . , st , and w e further define P := n p > st : p is prime and p | ( q + r D ) for some 0 < | q | < s, | r | < t o . Since there are only finitely man y pairs ( q , r ) with 0 < | q | < s and | r | < t , and for eac h such pair the integer q + r D is non-zero and hence has only finitely man y prime divisors, the set P is finite. No w, for eac h p ∈ P , consider the residue classes − i − j D (mo d p ) , where i and j run o ver the in tegers with 1 ≤ i ≤ s and 1 ≤ j ≤ t resp ectively . In particular, the n umber of residue classes − i − j D (mo d p ) is at most st . As p > st , these classes do not exhaust all residue classes mo dulo p , so that b y the Chinese Remainder Theorem w e deduce that there exists an in teger M > 2 s + 2 tD with M ≡ − i − j D (mo d p ) for all p ∈ P and all i and j . With this M , we define α i,j := M + i + j D and A := { α i,j : 1 ≤ i ≤ s, 1 ≤ j ≤ t } . T o see that the α i,j ∈ A are distinct (so that | A | = st ), w e note that α i,j = α k,l implies that D divides k − i . As | k − i | < s < D , we then get i = k , and consequently j = l . W e furthermore ha v e the following claim. ✓ ✓ Claim 3.2. F or any indic es ( i, j ) and ( k, l ) with 1 ≤ i, k ≤ s and 1 ≤ j, l ≤ t , one has gcd( α i,j , α k,l ) | ( i − k ) . Pr o of. Without loss of generalit y we may assume i  = k , as ev ery integer divides 0 . Now, with g := gcd( α i,j , α k,l ) and with v p ( · ) denoting the p -adic v aluation, we aim to pro ve that v p ( g ) ≤ v p ( i − k ) for all prime divisors p of g . Hence, let p b e an arbitrary prime divisor of g , and define q := i − k  = 0 and r := j − l . Since p | α i,j and α i,j − α k,l = q + r D , we hav e M ≡ − i − j D (mo d p ) and p | ( q + r D ) . Because 0 < | q | < s and | r | < t , if p > st , then the definition of P w ould imply that p ∈ P , con tradicting the choice of M . Hence p ≤ st , which w e claim implies that v p ( D ) > v p ( q ) . As p ≤ st and D = lcm(1 , 2 , . . . , st ) , we hav e v p ( D ) > 0 , so that the inequality v p ( D ) > v p ( q ) certainly holds if v p ( q ) = 0 . On the other hand, if v p ( q ) ≥ 1 , then we note that q 2 | D , as q 2 < s 2 ≤ st . So in this case we get v p ( D ) ≥ v p ( q 2 ) = 2 v p ( q ) > v p ( q ) , MA TCHING INTEGERS TO DISTINCT MUL TIPLES 5 where the final inequalit y uses q  = 0 . Th us indeed v p ( D ) > v p ( q ) in b oth cases, from whic h w e see v p ( q + r D ) = v p ( q ) . Since g | ( q + r D ) , we no w obtain v p ( g ) ≤ v p ( q + r D ) = v p ( q ) . As this holds for every prime p , w e deduce that g do es indeed divide q = i − k . □ By the generalized Chinese Remainder Theorem (see [12, Theorem 3.12]), an integer x 0 exists with x 0 ≡ i (mo d α i,j ) for all α i,j ∈ A , as long as, for all i, j, k , l , the difference of the residue i − k is divisible by gcd( α i,j , α k,l ) . As this is precisely the con ten t of Claim 3.2, suc h an x 0 do es indeed exist. W e no w define x := x 0 − M and I := ( x, x + 2 max A ) , and claim that for ev ery fixed pair ( i, j ) , there are at most t w o multiples of α i,j in I . T o see this, first note that the in teger x 0 − i is a multiple of α i,j , by construction of x 0 . As the m ultiples of α i,j form an arithmetic progression with common difference α i,j , it suffices to show that ( x 0 − i ) − α i,j < x and ( x 0 − i ) + 2 α i,j > x + 2 max A. Indeed, on the one hand, ( x 0 − i ) − α i,j = x + M − i − ( M + i + j D ) = x − 2 i − j D < x. On the other hand, ( x 0 − i ) + 2 α i,j = x + M − i + 2( M + i + j D ) = x + 3 M + i + 2 j D > x + 3 M > x + 2 M + 2 s + 2 tD = x + 2 max A. An y multiple of α i,j lying in I must therefore b e either x 0 − i or x 0 − i + α i,j = x 0 + M + j D. In particular, every multiple of an element of A that lies in I b elongs to { x 0 − i : 1 ≤ i ≤ s } ∪ { x 0 + M + j D : 1 ≤ j ≤ t } . As this latter set has at most s + t elements, this finishes the pro of. □ ✓ ✓ R emark 3.3 . Let ϵ ∈ (0 , 1) b e arbitrary and recall that w e c hose M > 2 s + 2 tD . By instead c ho osing M larger than ϵ − 1 (3 − ϵ )( s + tD ) , the ab o ve pro of works ev en if w e increase I to ( x, x + (3 − ϵ ) max A ) . 4. The lo wer bound W e now pro v e the low er b ound for f ( m ) . ✓ ✓ Theorem 4.1. F or every p ositive inte ger m , f ( m ) ≥ min( m, ⌈ 2 √ m ⌉ ) . Pr o of. Fix m ∈ N , and let A = { a 1 < · · · < a m } b e an y set of m p ositiv e integers. F urthermore, with x an arbitrary real n um b er, define the in terv al I := ( x, x + 2 a m ) . As explained in Section 2, with B := I ∩ Z and G the corresp onding bipartite graph with vertex sets A and B , it suffices to sho w inequality (2) for a fixed but arbitrary subset S ⊆ A . In order to do this, w e 6 W. V AN DOORN, Y. LI, AND Q. T ANG need to distinguish b etw een the case where x is an integer multiple of a m , and the case where it is not. Case 1: x / ∈ a m Z . W e consider the partition B = B − ⊔ B + with B − := ( x, x + a m ] ∩ Z , and B + := ( x + a m , x + 2 a m ) ∩ Z , and define Γ − ( S ) := Γ( S ) ∩ B − , Γ + ( S ) := Γ( S ) ∩ B + . Since Γ − ( S ) and Γ + ( S ) together partition Γ( S ) , w e note that | Γ( S ) | = | Γ − ( S ) | + | Γ + ( S ) | . (3) Moreo v er, we claim the inequalit y | S | ≤ | Γ − ( S ) | | Γ + ( S ) | . (4) T o see this, for each a ∈ S , let u a b e the largest multiple of a that is con tained in B − . As a is at most a m , u a certainly exists. Moreov er, b y definition of u a and the assumption that a m do es not divide x , w e ha v e that u a + a is a multiple of a contained in B + . Now consider the function ϕ : S → Γ − ( S ) × Γ + ( S ) , ϕ ( a ) := ( u a , u a + a ) . Since the difference of the tw o co ordinates is exactly a , this map is injective, whic h indeed giv es (4). Combining (3) and (4) with the AM–GM inequality already finishes the pro of for this case, as we then get | S | ≤ | Γ − ( S ) | | Γ + ( S ) | ≤ ( | Γ − ( S ) | + | Γ + ( S ) | ) 2 4 = | Γ( S ) | 2 4 , implying (2). Case 2: x ∈ a m Z . In this case w e define b 0 := x + a m , and let G 0 b e the subgraph of G with v ertex sets A 0 := A \ { a m } and B 0 := B \ { b 0 } . Let B − and B + b e defined as b efore, and define e B − := B − \ { b 0 } . F or S ⊆ A 0 , w e no w write Γ 0 ( S ) for its neigh b ourho o d in G 0 , and w e set e Γ − ( S ) := Γ 0 ( S ) ∩ e B − and e Γ + ( S ) := Γ 0 ( S ) ∩ B + . In analogy with Case 1 , it suffices to find an injective function ϕ : S → e Γ − ( S ) × e Γ + ( S ) for ev ery subset S ⊆ A 0 . Indeed, such a function would imply the analogous v ersion of (4), from which | Γ 0 ( S ) | ≥ 2 p | S | follo ws. W e w ould then b y Lemma 2.3 deduce the existence of a matc hing of size at least min( m − 1 , 2 √ m − 1) in G 0 , and adding the edge a m ∼ b 0 finishes the pro of, b y pro viding a matching of size at least min( m − 1 , 2 √ m − 1) + 1 ≥ min( m, 2 √ m ) in the original graph G . Hence, we fix a subset S ⊆ A 0 and, for each a ∈ S , w e let u a once again b e the largest m ultiple of a contained in B − . If a ∤ b 0 w e sa y that a is of type (T1), and we define ϕ ( a ) = ( u a , u a + a ) as before. If a do es divide b 0 , we remark that u a = b 0 . In this case, if w e furthermore hav e 2 a < a m and 2 a ∤ b 0 , then we sa y that a is of type (T2), and we define ϕ ( a ) = ( u a − 2 a, u a + 2 a ) . Finally , if a | b 0 but either 2 a ≥ a m or 2 a | b 0 , then w e refer to a as b eing of type (T3), and we define ϕ ( a ) = ( u a − a, u a + a ) . With these definitions it is quic kly v erified that ϕ ( S ) ⊆ e Γ − ( S ) × e Γ + ( S ) , so it suffices to show that ϕ is injective. MA TCHING INTEGERS TO DISTINCT MUL TIPLES 7 F or a start, as the difference of the coordinates is a , 4 a , or 2 a (for elements a of t yp e (T1), (T2), (T3) resp ectiv ely), we certainly hav e ϕ ( a )  = ϕ ( a ′ ) if distinct a, a ′ ∈ S are of the same t yp e. W e will therefore b y con tradiction assume that ϕ ( a ) = ϕ ( a ′ ) with a and a ′ of differen t t yp es, and note that this implies that at least one is a divisor of b 0 , so that b 0 is exactly the a v erage of the tw o co ordinates. First supp ose that a is of type (T1) and a ′ is of t yp e (T2). Then ϕ ( a ) = ϕ ( a ′ ) implies a = 4 a ′ b y the difference of their co ordinates. In particular, as u a is a m ultiple of a , u a m ust b e divisible by 2 a ′ . How ev er, this gives b 0 = u a + a 2 = 2 a ′  u a 2 a ′  + 2 a ′ = 2 a ′  u a 2 a ′ + 1  , con tradicting the fact that 2 a ′ ∤ b 0 as a ′ is of type (T2). Next, suppose that a is of type (T1) and a ′ is of t yp e (T3), in which case ϕ ( a ) = ϕ ( a ′ ) implies a = 2 a ′ . Since 2 a ′ = a < a m , this implies 2 a ′ | b 0 b y the assumption that a ′ is of type (T3). This also leads to a con tradiction, as b 0 = u a + a 2 = 2 a ′ · u a 2 a ′ + a ′ = a ′  2 · u a 2 a ′ + 1  , whic h is an o dd multiple of a ′ . Finally , supp ose that a is of t yp e (T2) and a ′ is of type (T3). Then ϕ ( a ) = ϕ ( a ′ ) implies a ′ = 2 a , while a ′ | b 0 as a ′ is of type (T3). But this con tradicts the fact that 2 a ∤ b 0 . The opp osite orderings are symmetric, so no mixed-type collision can o ccur. Hence ϕ is indeed injectiv e, finishing the pro of. □ 5. Lean f ormaliza tion W e used Aristotle, the automated theorem pro ving to ol from Harmonic, in order to obtain formal pro ofs of all results in this paper. One interesting feature of this formalization pro cess w as that w e discov ered, only after the formalization had already been completed, that the original proof draft pro duced b y ChatGPT con tained a gap. That gap w as subsequently repaired, and the present pap er reflects the corrected argument. More precisely , on closer insp ection it b ecame clear that ChatGPT’s initial prop osal for ϕ : S → e Γ − ( S ) × e Γ + ( S ) defined in Case 2 of the low er b ound, was not an injection. This w as of course worrying, but also confusing, as the formal pro of had already b een finished. W e then had to translate the Lean pro of back into informal language, to figure out what had happ ened. It then turned out that Aristotle had not only noticed the gap, but also managed to come up with a v ariation of ϕ that actually did work! Hence, Aristotle’s definition of ϕ is the one that we actually used in Section 4 in order to prov e Theorem 4.1. As for the accompanying Lean file [6], the file header records the following metadata: • Lean v ersion: leanprover/lean4:v4.28.0 ; • Mathlib v ersion: 8f9d9cff6bd728b17a24e163c9402775d 9e6a365 ; • The formalization w as obtained by Aristotle from Harmonic. The file consists of 1116 lines and contains 8 definitions, 42 lemmas, and 5 named theorems: • erdos_f_upper_bound , • large_3maxA_version , • erdos_f_lower_bound , • erdos_f_eq . • erdos_f_eq_ge4 . These latter 5 results refer to Theorem 3.1, Remark 3.3, Theorem 4.1, Theorem 2.1 and Remark 2.2 resp ectively . F or background on Lean and Mathlib, see [5, 13]. 8 W. V AN DOORN, Y. LI, AND Q. T ANG References [1] T. A chim et al., Aristotle: IMO-level Automate d The or em Pr oving , arXiv:2510.01346 [cs.AI], 2025. doi:10.48550/arXiv.2510.01346 [2] T. F. Blo om, Er dős Pr oblem #650 , https://www.erdosproblems.com/650 , accessed 2026-03-30. [3] T. F. Blo om, Er dős Pr oblem #650: discussion thr e ad , https://www.erdosproblems.com/forum/thread/ 650 , accessed 2026-03-30. [4] J. A. Bondy and U. S. R. Murty , Gr aph The ory , Graduate T exts in Mathematics, V ol. 244, Springer, New Y ork, 2008. [5] The mathlib Comm unity , The Lean mathematical library , in Pr o c e e dings of the 9th A CM SIGPLAN International Confer enc e on Certifie d Pr o gr ams and Pr o ofs (CPP 2020) , ACM, 2020, 367–381. [6] W. v an Doorn, Er dosPr oblem650.lean , GitHub repository , https://github.com/Woett/Lean- files/blob/ main/ErdosProblem650.lean , accessed 2026-03-30. [7] P . Erdős, Problems and results in combinatorial analysis and combinatorial num ber theory , in Pr o c e e dings of the Ninth Southe astern Confer enc e on Combinatorics, Gr aph The ory and Computing (Florida Atlan tic Univ., Bo ca Raton, Fla., 1978), Congr ess. Numer. XXI (1978), 29–40. [8] P . Erdős, Some problems on num b er theory , in Analytic and Elementary Numb er Theory (Marseille, 1983), Publ. Math. Orsay 86 (1986), 53–67. [9] P . Erdős, Some problems on num b er theory , in Octo gon Math. Mag. 3 (1995), 3–5. [10] P . Erdős and J. Surányi, Meg jegyzések egy versen yfeladathoz, Mat. Lap ok 10 (1959), 39–48. [11] Harmonic, Aristotle , https://aristotle.harmonic.fun/ , accessed 2026-03-30. [12] G. A. Jones and J. M. Jones, Elementary Numb er The ory , Springer Undergraduate Mathematics Series, Springer-V erlag London, London, 1998. [13] L. de Moura and S. Ullrich, The Lean 4 theorem prov er and programming language, in Automate d Deduc- tion — CADE 28 , LNAI 12699, Springer, 2021, 625–635. [14] Op enAI, GPT-5 , https://openai.com/gpt- 5/ , accessed 2026-03-30. [15] Q. T ang, A Note on Problem #650, GitHub rep ository file , a v ailable at https://github.com/QuanyuTang/ erdos- problem- 650/blob/main/650_lowerbound_v1.pdf . [16] Q. T ang, A Note on a Gap in Case 2 of the previous Low er-Bound Argument for Problem #650, GitHub r epository file , av ailable at https://github.com/QuanyuTang/erdos- problem- 650/blob/main/gap_in_ previous_proof.pdf . Gr oningen, the Netherlands Email addr ess : wonterman1@hotmail.com School of Ma thema tics, Southeast University, Nanjing 211189, P. R. China Email addr ess : liyanyang1219@gmail.com School of Ma thema tics and St a tistics, Xi’an Jiaotong University, Xi’an 710049, P. R. China Email addr ess : tang_quanyu@163.com