Almos i h powe s in a i hme ic p og ession
L. Hajdu and T. Ko ´acs
Uni e si y o Deb ecen, Ins i u e o Ma hema ics
and he Numbe Theo y Resea ch G oup o he Hunga ian Academy o Sciences
Deb ecen, Hunga y
Uni e si y o Deb ecen, Ins i u e o Ma hema ics
Deb ecen, Hunga y
Abs ac
We p o e ha he p oduc o kconsecu i e e ms o a p imi i e a i hme ic p og es-
sion is ne e a pe ec i h powe when 3 ≤k≤54. We also p o ide a mo e p ecise
s a emen , conce ning he case whe e he p oduc is an ”almos ” i h powe . Ou
heo ems yield conside able imp o emen s and ex ensions, in he i h powe case, o
ecen esul s due o Gy˝o y, Hajdu and Pin ´e . While he ea lie esul s ha e been
p o ed by classical (mainly algeb aic numbe heo e ical) me hods, ou p oo s a e
based upon a new ool: we apply genus 2 cu es and he Chabau y me hod (bo h
he classical and he ellip ic e ison).
Key wo ds: pe ec powe s, a i hme ic p og ession, genus 2 cu es, Chabau y
me hod
PACS: 11D41, 11B25
1 In oduc ion
Conside he Diophan ine equa ion
x(x+d). . . (x+ (k−1)d) = byn(1)
Email add esses: [email p o ec ed],[email p o ec ed] (L.
Hajdu and T. Ko ´acs).
1Resea ch suppo ed in pa by he Hunga ian Academy o Sciences, by he
OTKA g an s K67580, K75566, and by he T´
AMOP 4.2.1./B-09/1/KONV-2010-
0007 p ojec . The p ojec is implemen ed h ough he New Hunga y De elopmen
Plan, co- inanced by he Eu opean Social Fund and he Eu opean Regional De el-
opmen Fund.
in non-ze o in ege s x, d, k, b, y, n wi h gcd(x, d) = 1, d≥1, k≥3, n≥2 and
P(b)≤k. He e P(u) s ands o he la ges p ime di iso o a non-ze o in ege
u, wi h he con en ion P(±1) = 1.
The equa ion has a e y ich li e a u e. Fo d= 1 and b= 1, equa ion (1)
has been sol ed by E d˝os and Sel idge [9]. This celeb a ed esul can be
e o mula ed as ha he p oduc o wo o mo e consecu i e posi i e in ege s
is ne e a pe ec powe . The comple e solu ion o (1) in case o d= 1 is due
o Sa adha [21] (case k≥4) and Gy˝o y [10] (case k < 4).
Fo an o e iew o he huge numbe o ela ed esul s o d > 1 we e e o
su ey pape s o Gy˝o y [11], Sho ey [22], [23] and Tijdeman [25]. Now we
men ion only esul s which a e closely ela ed o he scope o he p esen
pape , ocusing on he comple e solu ion o (1) when he numbe ko e ms
is fixed.
In case o (k, n) = (3,2) equa ion (1) has infini ely many solu ions, al eady
o b= 1 (c. . [25]). Eule (see [8]) p o ed ha (1) has no solu ions wi h
b= 1, and (k, n) = (3,3) o (4,2). Obl´a h [18], [19] ob ained simila esul s
o (k, n) = (3,4), (3,5) and (5,2).
By a conjec u e o E d˝os, equa ion (1) has no solu ions in posi i e in ege s
when k > 3 and b= 1. In o he wo ds, he p oduc o kconsecu i e e ms o a
p imi i e posi i e a i hme ic p og ession wi h k > 3 is ne e a pe ec powe .
By p imi i e a i hme ic p og ession we mean one o he o m
x, x +d, . . . , x + (k−1)d,
wi h gcd(x, d) = 1. The conjec u e o E d˝os has ecen ly been e ified o
ce ain alues o kin a mo e gene al o m; see he pape s [11], [12], [1], [13].
Since now we ocus on he case n= 5, we gi e only he bes known esul
o his pa icula exponen . (Though he esul s men ioned a e alid o any
n≥2.) The ollowing s a emen is a combina ion o esul s om [11] (case
k= 3), [12] (cases k= 4,5), [1] (cases k= 6,7) and [13] (cases 8 ≤k≤34).
Theo em A. The only solu ions o equa ion (1) wi h n= 5,3≤k≤34 and
P(b)≤Pk, wi h
Pk=
2,i k= 3,4,
3,i k= 5,
5,i k= 6,7,
7,i 8≤k≤22,
k−1
2,i 23 ≤k≤34
a e gi en by
(k, d) = (8,1), x ∈ {−10,−9,−8,1,2,3}; (k, d) = (8,2), x ∈ {−9,−7,−5};
2
(k, d) = (9,1), x ∈ {−10,−9,1,2}; (k, d) = (9,2), x ∈ {−9,−7};
(k, d) = (10,1), x ∈ {−10,1}; (k, d, x) = (10,2,−9).
No e ha knowing he alues o k, d and x, all solu ions (x, d, k, b, y, n) o (1)
can be easily lis ed.
To explain why he case n= 5 in equa ion (1) is special, we need o gi e
some insigh in o he me hod o sol ing (1) o fixed k, in he gene al case
n≥2. One o he mos impo an ools is he modula me hod, de eloped by
Wiles [26]. In [11], [12], [1], [13] all h ee ypes o e na y equa ions (i.e. o
signa u es (n, n, 2),(n, n, 3),(n, n, n)) and ela ed esul s o Wiles [26], K aus
[16], Da mon and Me el [7], Ribe [20], Benne and Skinne [2], Benne ,
Va sal and Yazdani [3] and o he s a e used. Howe e , he modula echnique
wo ks effec i ely only o ”la ge” exponen s, ypically o n≥7. Thus he
”small” exponen s n= 2,3,5 mus be handled sepa a ely. In ac hese cases
a e conside ed in dis inc sec ions, o a e co e ed by sepa a e heo ems in he
abo e men ioned pape s.
Fu he , he exponen s n= 2,3 has al eady been conside ed in sepa a e pa-
pe s. Equa ion (1) wi h n= 2 has a b oad li e a u e in i sel ; see e.g. [15] and
he e e ences gi en he e. He e we ocus only on he esolu ion o (1) wi h
fixed k. Fo n= 2 and posi i e x, equa ion (1) has been comple ely sol ed (up
o a ew excep ional cases) by Hi a a-Kohno, Laish am, Sho ey and Tijdeman
[15] o k≤100, and in case o b= 1, e en o k≤109. Thei main ools we e
ellip ic cu es and quad a ic esidues. La e , he excep ional emaining cases
ha e been handled by Tengely [24], by he help o he Chabau y me hod. A
his poin we no e ha we shall e e o he Chabau y me hod equen ly in
his pape . Fo he desc ip ion o he me hod, and in pa icula how o use
i in he ame o he p og am package Magma [4], we e e o he pape s o
B uin [5], [6] and he e e ences gi en he e.
When n= 3, wo king mainly wi h cubic esidues, howe e making use o el-
lip ic cu es and he Chabau y me hod as well, Hajdu, Tengely and Tijdeman
[14] ob ained all solu ions o equa ion (1) wi h k < 32 such ha P(b)≤ki
4≤k≤12 and P(b)< k i k= 3 o k≥13. Fu he , i b= 1 hen hey could
sol e (1) o k < 39.
The case n= 5 has no ye been closely in es iga ed. In his case (in he abo e
men ioned pape s conside ing equa ion (1) o gene al exponen n) mainly
classical me hods we e used, due o Di ichle and Lebesgue (see e.g. [13]).
Appa en ly, o n= 5 ellip ic cu es a e no applicable. In he p esen pape
we show ha in his case he Chabau y me hod (bo h he classical and he
ellip ic e sion) can be applied e y efficien ly. As we men ioned, he Chabau y
me hod has been al eady used o he cases n= 2,3 in [1], [24], [14]. Howe e ,
3
i has been applied only o some pa icula cases and equa ions. To p o e ou
esul s we sol e a la ge numbe o genus 2 equa ions by Chabau y me hod,
and hen build a kind o sie e sys em based upon hem.
2 New esul s
Ou fi s heo em conside ably ex ends Theo em A, in he mos in e es ing
case o b= 1 in equa ion (1). We call an a i hme ic p og ession o he o m
x, x +d, . . . , x + (k−1)dp imi i e, i gcd (x, d) = 1.
Theo em 1 The p oduc o kconsecu i e non-ze o e ms in a p imi i e a i h-
me ic p og ession wi h 3≤k≤54 is ne e a fi h powe .
In ac Theo em 1 ollows di ec ly om he nex esul . To o mula e i , we
need o in oduce a new concep . An a i hme ic p og ession x, x +d, . . . , x +
(k−1)dis called i ial i d≤5 and |x+id| ≤ 15 o some i= 0,1, . . . , k −1.
Fu he , a solu ion o equa ion (1) is also called i ial, i he e ms x, x +
d, . . . , x+(k−1)don he le -hand side o (1) o m a i ial a i hme ic p og es-
sion. This concep is needed because o he huge numbe o i ial solu ions;
on he o he hand, such solu ions o (1) can be lis ed easily o any fixed k.
Theo em 2 Equa ion (1) wi h n= 5,3≤k≤24 and P(b)≤Pkhas he
only non i ial solu ions wi h
(k, d) = (3,7), x ∈ {−16,−8,−6,2};
(k, d) = (4,7), x ∈ {−16,−15,−12,−9,−6,−5};
(k, d) = (4,11), x ∈ {−27,−6}; (k, d) = (5,7), x ∈ {−16,−12};
(k, d) = (5,11), x ∈ {−36,−32,−12,−8};
(k, d) = (5,13), x ∈ {−40,−27,−25,−12};
(k, d) = (6,7), x ∈ {−32,−25,−10,−3};
(k, d) = (6,9), x ∈ {−25,−20}; (k, d) = (6,13), x ∈ {−40,−25};
(k, d) = (7,7), x ∈ {−39,−32,−27,−22,−20,−15,−10,−3};
(k, d) = (8,7), x ∈ {−39,−27,−22,−10};
(k, d) = (9,7), x ∈ {−39,−34,−32,−24,−22,−17};
(k, d) = (10,7), x ∈ {−39,−24},
4
whe e he alues o Pka e gi en by
k3 4 5 6 7,8
Pk3 5 7 11 13
k9,10,11,12 13,14,15 16,17 18,19,20,21,22,23 24
Pk17 19 23 29 31
Obse e ha Pk> k o k≥4 in Theo em 2, which is a new ea u e abou
equa ion (1).
As a simple and immedia e co olla y o Theo em 2 we ge he ollowing s a e-
men , conce ning he case P(b)≤k. We men ion ha al eady his esul
yields conside able imp o emen o Theo em A, in pa icula wi h espec o
he bound o P(b).
Co olla y 3 Fo n= 5 and 3≤k≤36 all non i ial solu ions o equa ion
(1) wi h P(b)≤ka e gi en by
(k, d) = (3,7), x ∈ {−16,−8,−6,2}; (k, d) = (5,7), x ∈ {−16,−12}.
Ou las heo em p o ides he key o he p oo o Theo em 2 in case o k≥4.
I has been p o ed by a kind o sie ing p ocedu e, based upon genus 2 equa-
ions and he Chabau y me hod. No e ha ha ing an inc easing a i hme ic
p og ession z1< . . . < zl, by symme y we ob ain ha −zl< . . . < −z1is
also an inc easing a i hme ic p og ession. Hence dealing wi h such a i hme ic
p og essions i is sufficien o gi e only one p og ession om each symme ic
pai .
Theo em 4 Le 4≤ ≤8and z0< z1< . . . < z −1be a non- i ial p imi i e
a i hme ic p og ession. Suppose ha
z0=b0x5
0, zi1=bi1x5
i1, zi2=bi2x5
i2, z −1=b −1x5
−1,
wi h some indices 0< i1< i2< −1such ha P(b0bi1bi2b −1)≤5. Then he
ini ial e m z0and common diffe ence z1−z0o he a i hme ic p og ession
z0, . . . , z −1 o he sepa a e alues o = 4, . . . , 8up o symme y is one o
= 4 : (−9,7),(−6,7),(−6,11),(−5,7);
= 5 : (−32,17),(−25,13),(−20,11),(−16,13),(−12,7),(−12,11),(−12,13),
(−10,7),(−8,7),(−8,11),(−4,7),(−3,7),(−1,7),(2,7),(4,7),(4,23);
5
= 6 : (−125,61),(−81,17),(−30,31),(−25,8),(−25,11),(−25,13),(−25,17),
(−20,9),(−20,13),(−20,19),(−20,29),(−15,7),(−15,11),(−15,13),(−15,23),
(−10,7),(−10,11),(−8,7),(−5,7),(−3,7),(−1,11),(−1,13),(1,7),(5,11);
= 7 : (−54,19),(−54,29),(−48,23),(−30,11),(−30,13),(−27,17),(−24,13),
(−18,7),(−18,11),(−18,13),(−18,19),(−16,11),(−15,7),(−12,7),(−12,11),
(−10,7),(−6,7),(−6,11),(−4,9),(−3,13),(−2,7),(−2,17),(2,13),(3,7),(6,7),
(8,7),(9,11),(18,7);
= 8 : (−405,131),(−125,41),(−100,49),(−32,11),(−27,11),(−27,13),
(−25,19),(−24,7),(−16,13),(−10,13),(−9,7),(−5,11),(−4,7),(−2,11),
(−1,13),(−1,7),(1,7),(3,11),(4,11),(5,7),(6,17).
3 P elimina ies
Be o e gi ing he p oo s o ou esul s, we explain some p inciples and ech-
niques which shall be used a he equen ly la e on. We p esen hese ools
sepa a ely because in his way he s uc u e o ou p oo s will be mo e ans-
pa en .
3.1 Reducing equa ion (1) o a i hme ic p og essions o ”almos ” fi h powe s
In a s anda d way, as gcd(x, d) = 1 and n= 5, any solu ion o equa ion (1)
can be w i en as
x+id =aix5
i(i= 0,1, . . . , k −1) (2)
whe e xiis a non-ze o in ege and aiis a fi h powe ee posi i e in ege
wi h P(ai)≤k. This obse a ion jus ifies he i le o he pape , as well: he
membe s o he a i hme ic p og ession x, x +d, . . . , x + (k−1)da e ”almos ”
n- h powe s.
3.2 Lis ing he possible coefficien uples
Suppose ha
ai1x5
i1< ai2x5
i2<··· < ai x5
i (3)
a e (no necessa ily consecu i e) nonze o e ms o a p imi i e a i hme ic
p og ession, wi h aijas in (2). In his subsec ion we explain a me hod o lis
all he possible coefficien - uples (ai1, ai2, . . . , ai ) co esponding o (3).
6
Obse e ha knowing aijis equi alen o knowing he exponen s νp(aij) o
he p imes p≤kin he ac o iza ion o aij. Take an a bi a y p ime p≤k
di iding one o he e ms aijx5
ij, and suppose ha ij0is such an index ha
νp(aij0x5
ij0)≥νp(aijx5
ij) o all j= 1, . . . , .
Since he a i hme ic p og ession is assumed o be p imi i e, one can easily
check ha hen o all j= 1, . . . , wi h j=j0we ha e νp(aijx5
ij) = νp(j−j0).
As we ha e νp(aij0)<5, we can simply lis all possibili ies o he exponen s
o he p ime pin he coefficien s ai1, ai2, . . . , ai . Then combining hese pos-
sibili ies o all p imes p≤k, we can lis all he possible coefficien - uples
(ai1, ai2, . . . , ai ) which may occu in (3).
3.3 Local es ing o coefficien uples
As we will see, some o he coefficien uples lis ed in he p e ious subsec ion
in ac canno occu as coefficien s o fi h powe s in a i hme ic p og essions.
In many cases his can be shown al eady modulo mwi h some app op ia e
choice o m. We shall use he moduli m= 11,25.
Le 0 ≤i1< i2<···< i ≤k−1 be indices, and conside a coefficien
- uple (ai1, ai2, . . . , ai ), which in ac we would like o exclude - ha is, we
would like o show ha no co esponding subsequence
ai1x5
i1, . . . , ai x5
i (4)
o any app op ia e a i hme ic p og ession exis s. Fo his pu pose, conside
(4) modulo m(wi h m= 11 o 25). Obse e ha o ha e such a sequence, we
should find app op ia e fi h powe s modulo m. We check all he possibili ies.
(Since we wo k wi h m= 11 and m= 25, he fi h powe s modulo ma e
only {0,±1}and {0,±1,±7}, espec i ely.) Obse e ha by cop imali y, we
know ha m|aij1, aij2yields ha m|j1−j2. I we find ha no fi h powe s
modulo mexis ha ing also he p e ious p ope y, hen he ac ual coefficien
uple (ai1, . . . , ai ) is no alid in he sense ha no unde lying subsequence (4)
exis s. We shall illus a e how o use his es la e on.
3.4 Reducing he p oblem o genus 2equa ions
We ound wo ways o ge access o genus 2 equa ions.
7
3.4.1 Reduc ion me hod I
Suppose ha a0x5
0, a1x5
1, a2x5
2is an a i hme ic p og ession wi h nonze o e ms,
and wi h common diffe ence d. Then we ha e
(a1x5
1)2−a0x5
0·a2x5
2=d2
which a e he subs i u ions X=−x0x2/x2
1,Y=d/x5
1and A=a0a2,B=a2
1
yields he genus 2 equa ion
AX5+B=Y2
in X, Y ∈Q.
3.4.2 Reduc ion me hod II
Suppose ha
aix5
i, ajx5
j, aux5
u, a x5
a e ou e ms o an a i hme ic p og ession. Then we ha e
(j−u)aix5
i+ (u−i)ajx5
j= (j−i)aux5
u
and
(j− )aix5
i+ ( −i)ajx5
j= (j−i)a x5
.
Mul iplying hese iden i ies we ge an equa ion o he o m
AX10 +BX5Y5+CY 10 =DZ5,(5)
whe e A= (j−u)(j− )a2
i,B= ((j−u)( −i) + (u−i)(j− ))aiaj,
C= (u−i)( −i)a2
j,D= (j−i)2aua and X=xi,Y=xj,Z=xux . Then
om (5) we can easily ge bo h genus 2 equa ions o e Q
A1Z5
1+B1=X2
1and A2Z5
2+B2=X2
2
wi h he no a ion A1= 4AD,B1=B2−4AC,X1= 2AX5/Y 5+B,Z1=
Z/Y 2and A2= 4CD,B2=B2−4AC,X2= 2CY 5/X5+B,Z2=Z/X2,
espec i ely.
The a ional poin s on he genus 2 cu es ob ained by bo h me hods (unde
sui able assump ions) can be de e mined by he Chabau y me hod. Then,
ollowing he co esponding subs i u ions backwa ds we can de e mine he
ac ual membe s o he o iginal a i hme ic p og essions.
No e ha in ac in case o k= 3 in he p oo o Theo em 2 we also use genus 1
cu es o e some numbe fields, which can be ea ed by he ellip ic Chabau y
8
me hod. Howe e , since hese a e pa icula cases, we do no include hem in
his ”gene al” discussion.
4 P oo s
We gi e he p oo s o ou esul s in a specific o de . Fi s we p o e he case
k= 3 o Theo em 2. We do so because his esul is needed in he p oo o
Theo em 4, which is he nex s ep. The la e esul gi es he key o de i e
Theo em 2 o k≥4. Then we con inue by p o ing he cases k≥4 o Theo em
2 and i s co olla y. Finally, we gi e he p oo o Theo em 1, which easily ollows
om Theo em 2.
In he p oo o case k= 3 o Theo em 2 we shall make use o wo lemmas.
The fi s one is due o Benne , B uin, Gy˝o y, Hajdu [1].
Lemma 5 Le Cbe a posi i e in ege wi h P(C)≤5. I he Diophan ine
equa ion
X5+Y5=CZ5
has solu ions in nonze o cop ime in ege s X, Y and Z, hen C= 2 and X=
Y=±1.
The second lemma is a esul o K aus [16].
Lemma 6 Le Aand Bbe cop ime posi i e in ege s wi h AB = 2α3β o
nonnega i e in ege s αand βwi h α≥4. Then he Diophan ine equa ion
AX5+BY 5=Z5
has no solu ions in cop ime nonze o in ege s X, Y and Z.
P oo o he case k= 3 o Theo em 2. Fi s lis all he possible coefficien
iples (a0, a1, a2) as in (2). This can be done by he me hod explained in Sub-
sec ion 3.2. Al oge he we ob ain 182 such iples. Obse e ha a2x5
2, a1x5
1, a0x5
0
is also an a i hme ic p og ession. Hence by symme y i is sufficien o con-
side hose 106 iples o which a0≤a2. (I will be clea om ou me hod
ha we can do so wi hou loss o gene ali y indeed.)
Clea ly, a0x5
0, a1x5
1, a2x5
2is also an a i hme ic p og ession modulo 11 and 25. So
we can es he coefficien iples modulo 11 and 25, as explained in Subsec ion
3.3. A e he modulo 11 es we a e le wi h 88 iples; o example (1,1,6)
ge s excluded by his me hod. The es modulo 25 excludes 6 mo e iples
(e.g. (1,4,3)), and we a e le wi h 82 ones.
9
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os, J. L. Sel idge,The p oduc o consecu i e in ege s is ne e a
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