Journal of Number Theory 222 (2021) 426–428
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Journal of Number Theory
www.elsevier.com/locate/jnt
Corrigendum
Corrigendum to “On a generalization of
a conjecture of Grosswald” [J. Number Theory 216 (2020) 216–241]
Pradipto Banerjee∗, Ranjan Bera
IndianInstituteofTechnology–Hyderabad,DepartmentofMathematics,Kandi, Sangareddy, Medak,Telangana502285,India
a r t i c l e i n f o a bs t r a c t
Article history:
Received15November2020 Receivedinrevisedform17 November2020
Accepted18November2020 Availableonline23December2020 CommunicatedbyS.J.Miller
MSC:
11R32 1C08 33C45 Keywords:
BesselPolynomials
GeneralizedLaguerrepolynomials Irreducibility
Galoisgroup
WeextendtheresultofLemma 4,[1] tothecasethate= 0 and= 1 whichwasmissingin[1] butusedintheproofof Theorem 1,[1].
©2020ElsevierInc.Allrightsreserved.
DOIoforiginalarticle:https://doi.org/10.1016/j.jnt.2020.02.013.
* Correspondingauthor.
E-mailaddresses:[email protected](P. Banerjee),[email protected](R. Bera).
https://doi.org/10.1016/j.jnt.2020.11.002
0022-314X/©2020ElsevierInc. Allrightsreserved.
P. Banerjee, R. Bera / Journal of Number Theory 222 (2021) 426–428 427
1. Thecorrection
Onpage15,[1], inCase (ii),itisclaimedinthefifthdisplaythatfore∈ {0,1},the followingholds.
1
p−1+ max
1
p−1, 1
p−2,2 log(2n+β) pslogp
> 1 k.
Thiswasobtainedbyarguing thatμe(g)≥1/k andthat μe(g)< 1
p−1+ max
1
p−1, 1
p−2,2 log(2n+β) pslogp
. (1)
Forthelatterestimate,wehadreferredtoLemma 4,[1].TheboundobtainedinLemma 4, [1], isvalidonlyfore≥.While,according to(b),Corollary 4,ife= 0,then= 1,i.e., e< .Thus,inorderforourargumentsto workincase (ii),wemustjustifythevalidity of(1) inLemma 4,[1],inthecasethate= 0 and= 1.Inthepresentnote,weachieve this.
Wefollow thenotationsof [1]. Inthecase underconsideration,e= 0 and = 1.By (b),Corollary 4[1],thisisthecaseifβ=−2.Weletpbeaprimefactorofn−=n−1 satisfyingp>2+|β|= 2+|β|,as requiredinLemma 4, [1].Weareto establishthat
μ0(g)< 1
p−1+ max
1
p−1, 1
p−2,2 log(2n+β) pslogp
. (2)
Werecallfrom[1] that
μe(g) =μe,p(g) = max
ν(b0)−ν(bj)
j :e < j ≤n
where g(x) = n
j=0bjxj and ν(bj) is the highest power of p that divides bj. From Lemma 4,[1],wealreadyhavethat
μ1(g)< 1
p−1+ max
1
p−1, 1
p−2,2 log(2n+β) pslogp
.
Thus,inordertoestablish(2),itwouldsufficetoshow that ν(b0)−ν(b1)≤0.
Next,werecallfrom[1] that bj=
n
j
(2n+β−j)!
(n+β)! . Thus,
428 P. Banerjee, R. Bera / Journal of Number Theory 222 (2021) 426–428
b0/b1=(2n+β)!
(n+β)!
(n+β)!
n(2n+β−1)! =2n+β n .
Therefore,ν(b0)−ν(b1)>0 impliesthatp|(2n+β).Also,asperourhypothesis,pdivides n−1.Thus,pdivides2n+β−2n+ 2=β+ 2.Sincep>2+|β|,itmustbethatβ+ 2= 0.
Butthis isacontradictionsince|β|=−2 inthis case.Ourassertionnowfollows.
References
[1]P.Banerjee,R.Bera,OnageneralizationofaconjectureofGrosswald,J.NumberTheory216(2020) 216–241.
