$o$

数理解析研究所講究録
第 708 巻 1989 年 51-62

On the product of the terms of a finite arithmetic progression

byR. Tijdeman(Leiden, The Netherlands)

Let$a,$$a,$$d$

and$k$$a+d,$ $a+2d,$$(k-1\}d)$be positive integers, $k\geq 3$ . We consider the arithmetic progression$\ldots$, $a+(k-1)d$ and in particular the product $\Delta=a(a+d)\ldots(a+$.

There are two circles of problems we shall consider:
I: What can be said about the greatest prime factorof distinct prime divisors$\omega(\Delta)$

of$\Delta$$P(\Delta)$

of$\Delta$

and the number?
$\iota$

II: Can$\Delta$

be an (almost) perfect power? Can each ofan l-th power for some$\ell\geq 2$$a,$$a+d,$ $\ldots,$ $a+(k-1)d$ be?

This lexture reports on joint work with T.N. Shorey. It can be considered as an
updating of my first lecture given in Banff in 1988, [20]. Almost all results haveeffective proofs, but for this aspect I refer to the original papers. I am grateful toShorey for his remarks on an earlier draft of the present paper.

I. Without loss of generality we may assume $gcd(a, d)=1$ . A first general result on1

I was obtained by Sylvester [19] in 1892. He proved

(1)

if

$a\geq d+k$then $P(\Delta)>k$ .

Suppose $d=1$ . Then we consider the product of a block ofthen there is apparently at least one number in the block

which is not composed of primes$\leq k$$k$

integers. If $a\geq 1+k$ ,$a,$$a+1,$$\ldots$, $a+k-1$. If $a=1+k$ , this is Bertrand’s Postulate. If$a<1+k$ , the last term of the block is less than $2k$ . Then the question becomes
whether $k+1,$ $\ldots,a+k-1$ contains a prime. This is the classical problem on gaps
between consecutive primes. The theorem of Hoheisel, Ingham as improved by manyothers says that if $a>k^{23/42}$ and$k$is sufficiently large then there is a prime. (Theexponent 23/42 has been slightly improved, see [7]). Probably $a>(1+\epsilon)(\log k)^{2}$for large$k$is sufficient according to a hypothesis of Cram\’er, but it will be extremely

hard to prove this.
Suppose $d>1$ . Then (1) was slightly improved by Langevin in 1977 as follows.

lf $a>k$ ,(2)then $P(\Delta)>k$ .

Shorey and I showed that in fact 2, 9, 16 is the only exception:

Theorem 1. ([14]). Let $d>1,$ $k>2,gcd(a, d)=1,$ $(a, d, k)\neq(2,7,3)$ . Then$P(\Delta)>k$.

. The proof rests on a sharp upper bound for $\pi(x)$ due to Rosser and Schoenfeld andis further computational.
253lf$a$becomes large, then much better lower bounds are possible. Shorey and I

improved upon some estimate of Langevin [5].

Theorem 2. ((a) and (c) in [15], (b) unpublished). Let $\chi=a+(k-1)d,$$\max(\chi/k, 3)$$\epsilon>0$$\chi_{0}=$

,

.

(a)

$P(\Delta)>>k\log\log\chi_{0}$

(b)

if

$\chi>k(\log k)^{\epsilon}$

(c)

if

$\chi>k^{1+\epsilon}$

thenthen$(\geq k\log\log d)$$P(\Delta)>>\epsilon k$.

log log$P(\Delta)>>\epsilon k\log\log\chi$$a$.

.

The proof is based on Baker’s method, in particular a result on the Thue equation
by $Gy6ry[4]$ . Note that some conditions in (b) and (c) are necessary. $h(b)$ we

can take $a=[k/2],d=1$ and it follows that $P(\Delta)\leq a+(k-1)d<3k/2=$$o(k\log\log a)$. $h(c)$ we can take $a=1,$$d=[(\log\log\chi)^{1/2}]and^{\sim}$$P(\Delta)\leq a+(k-1)d<k(\log\log\chi)^{1/2}=o$ ( $k$ log log

Very recently we studied$\omega\backslash (\Delta)$. If $a=d=1$ , then$\chi$it follows that).

$\omega(\Delta)=\omega(k!)=\pi(k)$. There are

more examples with $\omega(\Delta)=\pi(k)$ , for example 1, 625, 1249, and 1, 3, 5, 7, 9.
Theorem 3. [16]$\omega(\Delta)\geq\pi(k)$.

The proof is sinilar to that of Theorem 1. Here only limited improvement is possible3

if$a$

becomes large. We cannot exclude that there are $k-l$ primes

such a way that 1,$p_{1},$$\ldots,p_{k-1}$$p_{1},$$\ldots,p_{k-1}$

inare in arithmetic progression, so that we cannot

prove anything better than $\omega(\Delta)\geq k-1$ .

Theorem 4. (a) For any positive integer $t>1$ we $have$
ifthen$\chi\geq k^{\frac{k}{\ell-1}+1}$(b) there are in finitely many instances with$\omega(\Delta)>k-t$$\chi\geq k^{2.7}$.

and $\omega(\Delta)<ck$ with $c<1$ .

The proof of (a) is elementary. For (b) we use estimates for the Dickman function$\psi(x, y)$. If we take $t=.6k$ in (a), then we find that $\omega(\Delta)>4k$ ifshows that $ck$ for someII. How many$\ell th$$c$$\chi\geq k^{2.7}$and (b)with $0<c<1$ is the actual order of magnitude.

powers can be in arithmetic progression? If $\ell=2$ , then there areinfinitely many triples of squares in arithmetic progression, but Fermat proved that
there are no four squares in arithmetic progression. D\’enes [1] proved in 1952 that

there are no three -th powers in arithmetic progression for $3\leq\ell\leq 30$ and for 60$\ell$other prime values of and he conjectured that this is true for all . He used Kummer$\ell$$\ell$

theory and his method is not applicable for irregular primes. The celebrated resultof Faltings [3] implies that for any$\ell\geq 5$

there are only finitely many triples ofcoprime -th powers in arithmetic progression, but his result does not provide any$\ell$

bound for$\ell\geq 2$

. Let

$k$

$d_{1}$

independent of$\ell$. In the sequel we assume $gcd(a, d)=1,$ $k\geq 3$ andbe the maximal divisor of$d$4

composed of prime factors $\equiv 1(mod l)$ .

$J^{\circ 0}$

Theorem 5. ((a), (c) and (d) from [12], (b) from [13]).
Suppose $a,a+d,$, $a+(k\cdot-1)d$ are all -th $po$ wers. Then$\ell$$\ldots$

is odd, then $k=3$ ,

(a)

if(b)

$k<(1+\epsilon)2^{\iota u(d_{1})}$(c)$k<<\omega(d)\log\omega(d)$(d)$k<<\sqrt{\log d}$

$d$

for

$k\geq k_{0}(\epsilon)$,,.

A much weaker condition is that not each of the numbers is an -th power, but that$\ell$

the product of the numbers is an -th power. If $d=1$ we find that$\ell$(3)$a(a+1)(a+2)\ldots(a+k-1)=y^{p}$$(\ell>1)$.

It was proved by Erd\"os and Selfridge [2] in 1975, 36 years after Erdos started thisresearch, that (3) has no solution in positive integers $a,$ $k>2,$ $y,$ $\ell>1$ . Later Erdos$t$conjectured that if(4)then$a(a+d)(a+2d)\ldots(a+(k-1)d)=y^{p}$$k$

is bounded by an absolute constant. Still later he conjectured $k\leq 3$ . Some

special cases are in the literature. Euler proved that the product of four numbers inarithmetic progression cannot be a square. Of course, this implies Fermat’s resultthat there are no four squares in arithmetic progression. Obl\’ath [8] proved the result
for the product of five numbers in arithmetic progression. He [9] also proved that5the product of three numbers in arithmetic progression cannot be a third, fourthof fifth power. Marszalek [6] was the first to deal with the general problem. He
proved that$k$is bounded by a number depending only on . He gave rather refined$d$estimates, but a rough simplification of his result gives:$k\leq\exp(2d^{3/2})$$k\leq\exp(2d^{7/3})$$k\leq Cd^{5/2}$$k\leq Cd$

Shorey [10] proved that$k$,,$\ell=4$ ,, whereifififif$\ell=2$$\ell=3$$C=3\cdot 10^{4}$$\ell\geq 5$.

is bounded by a number depending on $P(d)$ , the greatest

prime factor of , provided that $\ell>2$ . Shorey [10] further proved that $d_{1}>1$ if$d$$m>k$ and$k$large.

Shorey and I have obtained many results on equation (4). Actually we proved theseresults under the following weaker assumption.
(5)$\{\begin{array}{l}Leta,d,k,b,y,\ell bepositiveintegerssuchthatgcd(a,d)=1k>2,\ell>1.,.P(b)\leq k,P(y)>kanda(a+d).(a+(k-1)d)=by^{\ell}\end{array}$In the sequel we assume that (5) holds.

Theorem 6. $\log k<<\underline{\log d_{1}}$$\log\log d_{1}$Proof. For $P\geq 7$ , see [18]. For$\ell\leq 5$

see [17], formula (2.14).

Observe that this is a considerable improvement of Marszalek’s result. We see that$k/darrow 0$

as$darrow\infty$

Theorem 7. [17]and even $\log k/\log darrow 0$ as$k<<d_{1}^{1/(l-2)}$$darrow\infty$.

.

This implies Shorey’s estimate $d_{1}>1$ for $\ell>2$ and6$k$

large.

$l$

Theorem 8. [17]$\mathfrak{l}$$\frac{k}{\log k}<<\ell^{\omega\langle d)}$Thus$k$(even$<<\ell^{\omega(d_{1})}$is bounded by a number depending only on

to prove that

$k$

$p$

for

$\ell\geq 7$and).

$\omega(d)$is bounded by a number depending only on. Actually we tried$\omega(d)$, but we did not

succeed.
Suppose $p>2$ and $P(d)$ is bounded. Then$d_{1}$

is bounded or every prime factor ofis bounded by Theorem 7. However, by definition every prime factor oflarger than
$k$

$k$

$p$

. So we obtain that$\omega(d)$and$\ell$$d_{1}$isare bounded, hence, by Theorem 8,is bounded. Thus Theorems 7 and 8 generalize the results of Shorey mentioned

above.

Theorems 7 and 8 imply a slightly weaker inequality than Theorem 6 gives. It

is well known that $\omega(.n)<<\log$ n/log log$P\leq\log\log k/\log$log log$k$$n$for all $n>e$ . Suppose $\ell\geq 7$ . Ifthen Theorem 8 implies$\log k<<\frac{\log d}{\log\log d_{1}}$and if $P>\log\log$ k/log log log$k$log log log$k$then Theorem 7 implies$\log k<<\frac{\log d_{1}}{\ell}<\log d_{1}\frac{\log log\log k}{\log\log k}$.

Theorems 7 and 8 have proofs based on multiple application of the box principle.

For $P=2$ the proof is elementary, but complicated. For $P\geq 3$ the proof is completely7

different. For $p\geq 7$ we obtain the best results via elementary arguments, but for$P=3$ and 5 we reach the best estinates when we use Brun’s sieve and some resultof Evertse on the number of solutions of the equation $ax^{\ell}-by^{\ell}=c$ , proved by using
hypergeometric functions. I want to stress that many lemnas and arguments are

due to$Erd_{0}^{\infty}s$.

We have proved that $P(d)$ , and even $P(d_{1})$ , tends to$\infty$when $karrow\infty$ . In fact we

can prove
Theorem 9. [18]$P(d_{1})>>P\log k$ log log $k$for $P\geq 7$ ,$P(d)>>l\log k\log\log k$

for $p\in\{2,3,5\}$ .

$h[18]$ we give also lower bounds for the smallest prime factor and the greatestsquare free divisor of$d_{1}$.

Up to now I have restricted myself to dependence on$d,$ $d_{1},$ $k$

andcan also been taken into account.
Theorem 10 [17].
(a) There is an absolute constant$\ell_{0}$such that for$k<<a1$ .

(b)$k<<a,\{v(d)1$8.

$p\geq\ell_{0}$we have$\ell$. Of course,

$a$

$S$

For (a) see Shorey [11]. Further (b) follows from the combination of (a) and Theorem
8.

The last theorem concerns upper bounds for the largest term in the arithmetic

progression.
Theorem 11. [17]. There is an absol $ute$ constant$k_{0}$

such th at$a+(k-1)d\leq 17d^{2}k(\log k)^{4}$

if

$P=2$

$a+(k-1)d<<k( \frac{d}{\ell})^{p/(p-2)}$

if$\ell>2$$k\geq k_{0}$impliesand.

Finally I want to state a conjecture for the general situation in the line of theconjectures of D\’enes and Erdos stated above.

Coniecture. If (5) holds, then $k+P\leq 6$ .
If $k+\ell\leq 6$ , then $(k,\ell)=(3,3)$ or $(4, 2)$ . It is shown in [20] that in these cases there

are infinitely many solutions. As a more moderate target I challenge the reader toprove that$k$is bounded by a function of only$\omega(d)$if (5) is satisfied.

Mathematical Institute R.U.

Postbox 9512

2300 RA Leiden
The Netherlands.
9$\mathfrak{d}$俺References

1. P. D\’enes, Uber die diophantische Gleichung$x^{\ell}+y^{\ell}=cz^{\ell}$, Acta Math. 88(1952),

241-251.
2. P. $Erd6s$ and J.L. Selfridge, The product of consecutive integers is never apower, Illinois J. Math. 19(1975), 292-301.
3. G. Faltings, Endlichkeitss\"atze fur abelsche Varietaten \"uber $Zahlk\tilde{o}rpern$ , In-vent.

Math. 73(1983), 349-366.

4. K. Gyory, Explicit upper bounds for the solutions of some diophantine equations, Ann. Acad. Sci. Fenn. Ser. AI5 (1980), 3-12.
5. M. Langevin, Facteurs premiers d’entiers en progression arithm\’etique, Acta

Arith. 39(1981), 241-249.
6. R. Marszalek, On the product of consecutive elements of an arithmetic progression, Monatsh. Math. 100(1985), 215-222.

7. C.J. Mozzochi, On the difference between consecutive primes, J. Number Theory 24(1986), 181-187.

8. R. Obl\’ath, Uber das Produkt f\"unf aufeinander folgender Zahlen in einerarithmetischen Reihe, Publ. Math. Debrecen 1(1950), 222-226.
9. R. Oblath, Eine Bemerkung \"uber Produkte aufeinander folgender Zahlen, J.

hdian Math. Soc. (N.S.) 15(1951), 135-139.
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10. T.N. Shorey, Some exponential diophantine equations, New Advances in Transcendence Theory, ed. by A. Baker, Cambridge University Press, 1988, pp.

352-365.
11. T.N. Shorey, Some exponential diophantine equations II, to appear in Proceedings Bombay Intern. Colloquium held in January 1988.

12. T.N. Shorey and R. Tijdenan, Perfect powers in arithmetical progression,
Madras University J., to appear.
13. T.N. Shorey and R. Tijdeman, Perfect powers in arithmetical progression (II),in preparation.
14. T.N. Shorey and R. Tijdeman, On the greatest prime factor of an arithmeticalprogression, to appear.
15. T.N. Shorey and R. Tijdeman, On the greatest prime factor of an arithmeticalprogression (II), Acta Arith., to appear.

16. T.N. Shorey and R. Tijdeman, On the number of prime factors of an arith-metical progression, J. Sichuan Univ., to appear.

17. T.N. Shorey and R. Tijdeman, Perfect powers in products of terms in an arithmetical progression, to appear.
18. T.N. Shorey and R. Tijdeman, Perfect powers in products of terms in an arithmetical progression (II), in preparation.

19. J.J. Sylvester, On arithmetic series, Messenger Math. 21(1892), 1-19 and 87120.
11b2

20. R. Tijdeman, Diophantine equations and diophantine approximations, 1st Lecture, in: Number Theory and Applications, Ed. by R.A. Mollin, Kluwer,Dordrecht etc., 1989, pp.215-223.

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