Problem 672

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claims/: The 8 claim pages of Problem 672, one per claimant's result; the problem's standing derives from them.

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Statement. Can the product of an arithmetic progression of positive integers $n,n+d,\ldots,n+(k-1)d$ of length $k\geq 4$ (with $(n,d)=1$) be a perfect power?

Status. Verifiable, in the site's label (VERIFIABLE), an open label; refereed partial results settle instances in the negative (Current assessment).

Source. erdosproblems.com/672, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #672, https://www.erdosproblems.com/672.

References.

• [BBGH06] Bennett, M. A. and Bruin, N. and Győry, K. and Hajdu, L., Powers from products of consecutive terms in arithmetic progression. Proc. London Math. Soc. (3) (2006), 273-306.
• [BeSi20] Bennett, Michael A. and Siksek, Samir, A conjecture of Erdős, supersingular primes and short character sums. Ann. of Math. (2) (2020), 355-392.
• [ErSe75] Erdős, P. and Selfridge, J. L., The product of consecutive integers is never a power. Illinois J. Math. (1975), 292-301.
• [GHP09] Győry, K. and Hajdu, L. and Pintér, Á., Perfect powers from products of consecutive terms in arithmetic progression. Compos. Math. (2009), 845-864.
• [GHS04] Győry, K. and Hajdu, L. and Saradha, N., On the Diophantine equation $n(n+d)\cdots(n+(k-1)d)=by^l$. Canad. Math. Bull. (2004), 373-388.
• [Ma85] Marszałek, R., On the product of consecutive elements of an arithmetic progression. Monatsh. Math. (1985), 215-222.
• [Ob51] Oblath, Richard, Eine Bemerkung über Produkte aufeinander folgender Zahlen. J. Indian Math. Soc. (N.S.) (1951), 135-139. The site's commentary credits the case $(k,\ell)=(5,2)$ to Obláth under this key, which the site's reference record resolves to this 1951 note; Győry, Hajdu and Saradha (2004, reference [11]) and Bennett, Bruin, Győry and Hajdu (2006, references [26] and [27]) place that case in Obláth's earlier paper, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe, Publ. Math. Debrecen 1 (1950), 222-226, doi:10.5486/PMD.1950.1.2-4.29, the paper the commentary describes.

Formalization. Statement in formal-conjectures, whose variant erdos_672.variants.euler ($k=4$, $\ell=2$) carries a formal_proof link to a Lean development this repository has not built; see its claim page.

Current assessment

Here a primitive progression means $\gcd(n,d)=1$, with $n,d$ positive; it does not mean that every pair of terms is coprime. A perfect power means $y^\ell$ with an integer exponent $\ell\geq2$. The full question remains open in general on the catalog as accessed and the searches. It is verifiable in the site's sense: a single product that is a perfect power would settle it, while a negative answer needs a proof. Search scope: the primary sources and the site discussion, for a full resolution of the coprime positive-progression question; the Annals and arXiv versions of Bennett–Siksek's Theorem 2, with the Annals publisher record, for its prime-exponent restriction and finiteness clause; the published Győry–Hajdu–Pintér paper for the exact $4\leq k\leq34$ range; and the exact hypotheses of the 1975, 2004 and 2006 theorems against the site summary. The partial exclusions stand as recorded below, and no full resolution was found. The 2025 publisher description of Saradha Natarajan's Perfect Powers—An Ode to Erdős, under “About this book,” also describes the arithmetic-progression conjecture as unsolved (DOI 10.1007/978-981-96-2599-4). This is currentness corroboration, not primary theorem evidence.

The claim pages record the refereed partial results, each an accepted partial claim that settles instances in the negative: Obláth ($k=5$, $\ell=2$), Erdős–Selfridge ($d=1$), Marszałek ($k$ large in terms of $d$), Győry–Hajdu–Saradha ($k=4,5$), Bennett–Bruin–Győry–Hajdu ($4\le k\le11$), Győry–Hajdu–Pintér ($4\le k\le34$) and Bennett–Siksek ($k\ge k_0$ with a large prime exponent). The site credits Euler with the case $(k,\ell)=(4,2)$ without naming a publication, so that credit has no claim page of its own; the case lies inside the Győry–Hajdu–Saradha theorem, and a Lean proof of it posted in September 2026 is a pending claim, Piscitelli.

The proofs of these theorems are not reproduced here.

Known Results

The strongest length range recorded here is the published Győry–Hajdu–Pintér Theorem 1.1, Compositio Mathematica 145 (2009), 845–864, DOI 10.1112/S0010437X09004114. The theorem is on printed p. 847. The abstract on printed p. 845 and definition on p. 846 specify positive initial term and common difference with their gcd equal to one. It excludes perfect powers for every $4\leq k\leq34$ and arbitrary positive $d$ under those hypotheses. This is a partial length range, not a solution for all $k$ (its claim page).

Erdős–Selfridge (1975), Theorem 1 on printed p. 292, proves that no product of at least two consecutive positive integers is a perfect power, covering $d=1$ (its claim page). Section 4, printed p. 300, states an unnumbered assertion that for each fixed positive $d$ a $d$-dependent threshold $t_d$ excludes longer products. The paper gives no proof (it says "there must be" such a $t_d$) and no threshold uniform in $d$; it also notes infinitely many square examples of length three. Marszałek (1985) proved such a threshold, explicit in $d$ (its claim page), and Obláth (1950) had excluded squares for $k=5$ (its claim page).

For the more general equation

$$n(n+d)\cdots(n+(k-1)d)=b y^\ell,$$

Győry–Hajdu–Saradha (2004) uses positive integers $n,d,y,b$, integers $k,\ell\geq2$, $\gcd(n,d)=1$, $P(b)\leq k$, and $b$ free of $\ell$th powers, where $P(1)=1$. These ambient hypotheses appear on printed p. 373. Theorem 1, printed p. 374, states the exclusions for $k=4,5$ and $b=1$. Theorem 6, printed p. 375, gives finiteness in $n,d,b,y$ for fixed $k\geq3$, $\ell\geq2$ with $k+\ell>6$. The following remark describes infinitely many solutions in the complementary cases $k+\ell\leq6$ under those lower bounds. The later Bennett–Bruin–Győry–Hajdu paper, printed p. 273, explicitly identifies an invalid argument for $\ell=3$ in the 2004 paper and says it is corrected in its Section 5. The stated result stands with that corrected proof (its claim page).

Bennett–Bruin–Győry–Hajdu (2006), Theorem 1.1 on printed p. 274, excludes perfect powers for $4\leq k\leq11$ and arbitrary positive $d$ in a primitive positive progression (its claim page). Its further results concern finiteness, not exclusion of every solution. For its equation (5) above, $P$ denotes the largest prime divisor, with $P(\pm1)=1$.

• Theorem 1.4, printed pp. 275–276, gives at most finitely many solutions in nonzero integers $n,d,\ell,b,y$ for $12\leq k\leq82$, $\gcd(n,d)=1$, $\ell\geq2$, and $P(b)<k/2$. Every such solution satisfies $\log P(\ell)<3^k$.
• Theorem 1.5, printed p. 276, fixes $k\geq4$ and gives at most finitely many positive-integer solutions $n,d,b,y,\ell$ with $\gcd(n,d)=1$, $y>1$, $\ell>1$, $P(b)<k/2$, and $d\not\equiv0\pmod{D_k}$, where $D_k=\prod_{k/2\leq p<k,\ p\text{ prime}}p$. Every such solution satisfies $\log P(\ell)<3^k$. The restriction on $d$ is part of the theorem.
• Corollary 1.6, on the same page, fixes a positive integer $D$ and a length $k\geq4$ when $D=1,2$, or $k\geq6D\log D$ when $D\geq3$. It gives at most finitely many positive-integer solutions $n,d,b,y,\ell$ with $\gcd(n,d)=1$, $y>1$, $\ell>1$, $\omega(d)\leq D$, and $P(b)<k/2$, where $\omega$ counts distinct prime factors.

Bennett–Siksek, A conjecture of Erdős, supersingular primes and short character sums, Annals of Mathematics 191 (2020), 355–392, has a different conclusion. Published Theorem 2, printed p. 357, gives an effectively computable absolute $k_0$ such that for any fixed positive $k\geq k_0$, an integer solution of equation (2) with $\gcd(n,d)=1$ and prime exponent $\ell$ satisfies

$$y=0\quad\text{or}\quad d=0\quad\text{or}\quad \ell\leq\exp(10^k).$$

The following sentence invokes Faltings for finiteness; the abstract on printed p. 355 confirms at most finitely many positive solutions $n,d,y,\ell$, $\ell\geq2$, for each sufficiently large fixed $k$. This does not assert nonexistence for large $k$ in general; it excludes, for each $k\ge k_0$, every prime exponent $\ell>\exp(10^k)$ (its claim page).

The proofs of these theorems, including the 2006 correction and the 2020 finiteness deduction, are not reproduced here.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• bennett_2006_powers_products_consecutive_terms_arithmetic_progression
• bennett_2006_powers_products_consecutive_terms_arithmetic_progression / corollary_1_6
• bennett_2006_powers_products_consecutive_terms_arithmetic_progression / theorem_1_1
• bennett_2006_powers_products_consecutive_terms_arithmetic_progression / theorem_1_2
• bennett_2006_powers_products_consecutive_terms_arithmetic_progression / theorem_1_4
• bennett_2006_powers_products_consecutive_terms_arithmetic_progression / theorem_1_5
• bennett_2020_conjecture_erdos_supersingular_primes_short
• bennett_2020_conjecture_erdos_supersingular_primes_short / theorem_2
• erdos_1975_product_consecutive_integers_is_never_power
• erdos_1975_product_consecutive_integers_is_never_power / remark_p300
• erdos_1975_product_consecutive_integers_is_never_power / theorem_1
• erdos_1975_product_consecutive_integers_is_never_power / theorem_2
• gyory_2004_diophantine_equation
• gyory_2004_diophantine_equation / theorem_1
• gyory_2004_diophantine_equation / theorem_2
• gyory_2004_diophantine_equation / theorem_6
• gyory_2004_diophantine_equation / theorem_7
• gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression
• gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression / corollary_1_1
• gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression / theorem_1_1
• gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression / theorem_1_2
• gyory_2009_perfect_powers_products_consecutive_terms_arithmetic_progression / theorem_1_3