Let 0  be the numerical
semigroup generated by a and b with Frobenius number g(a,b)=ab-a-b. In this
note, we prove that there exists a prime number p in  with p < g(a,b) when
the product ab is sufficiently large. Two related conjectures are posed and
discussed as well

J.L. Ramírez Alfonsín

M. Skałba

Comptes Rendus Mathématique

arXiv

Let $0Frobenius number $g(a,b)=ab-a-b$. In this note, we prove that there exists a prime number $p\in \langle a,b\rangle $ with $

Ramírez Alfonsín, J.L.

Skałba, M.

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Comptes RendusMathématiqueJ.L. Ramírez Alfonsín and M. SkałbaPrimes in numerical semigroupsVolume 358, issue 9-10 (2020), p. 1001-1004Published online: 5 January 2021https://doi.org/10.5802/crmath.104This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orge-ISSN : 1778-3569Comptes RendusMathématique2020, 358, n 9-10, p. 1001-1004https://doi.org/10.5802/crmath.104Number Theory / Théorie des nombresPrimes in numerical semigroupsJ.L. Ramírez Alfonsín∗, a and M. Skałbaba UMI2924 - Jean- Christophe Yoccoz, CNRS-IMPA, Brazil and Univ. Montpellier,CNRS, Montpellier, Franceb Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa,PolandE-mails: jorge.ramirez-alfonsin@umontpellier.fr (J.L. Ramírez Alfonsín),skalba@mimuw.edu.pl (M. Skałba)Abstract. Let 0 < a < b be two relatively prime integers and let 〈a,b〉 be the numerical semigroup generatedby a and b with Frobenius number g (a,b) = ab−a−b. In this note, we prove that there exists a prime numberp ∈ 〈a,b〉 with p < g (a,b) when the product ab is sufficiently large. Two related conjectures are posed anddiscussed as well.2020 Mathematics Subject Classification. 11D07, 11N13.Funding. J.L. Ramírez Alfonsín was partially supported by Grant MATHAM-SUD 18-MATH-01, ProjectFLaNASAGraTA and INSMI-CNRS..Manuscript received 22nd July 2020, revised and accepted 30th July 2020.Let 0 < a < b be two relatively prime integers. Let S = 〈a,b〉 = {n |n = ax +by, x, y ∈Z, x, y ≥ 0}be the numerical semigroup generated by a and b. A well-known result due to Sylvester [5] statesthat the largest integer not belonging to S, denoted by g (a,b), is given by ab − a − b. g (a,b) iscalled the Frobenius number (we refer the reader to [3] for an extensive literature on the Frobeniusnumber).We clearly have that any prime p larger than g (a,b) belongs to 〈a,b〉. A less obvious and moreintriguing question is whether there is a prime p ≤ g (a,b) belonging to 〈a,b〉.In this note, we show that there always exists a prime p ∈ 〈a,b〉, p < g (a,b) when the productab is sufficiently large. The latter is a straight forward consequence of the below Theorem.Let 0 < u < v be integers. We defineπS [u, v] = |{p prime | p ∈ S, u ≤ p ≤ v}|.For short, we may write πS instead of πS [0, g (a,b)].Theorem 1. Let 3 ≤ a < b be two relatively prime integers and let S = 〈a,b〉 be the numericalsemigroup generated by a and b. Then, for any fixed ε> 0 there exists C (ε) > 0 such thatπS >C (ε) g (a,b)log(g (a,b))2+ε∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/1002 J.L. Ramírez Alfonsín and M. Skałbafor ab sufficiently large.Let us quickly introduce some notation and recall some facts needed for the proof of Theo-rem 1.Let S = 〈a,b〉 and let 0 < u < v be integers. We definenS [u, v] = |{n ∈N | u ≤ n ≤ v, n ∈ S}|andncS [u, v] = |{n ∈N | u ≤ n ≤ v, n 6∈ S}|.For short, we may write nS instead of nS [0, g (a,b)] and ncS instead of ncS [0, g (a,b)]. The set ofelements in ncS =N\ S are usually called the gaps of S.It is known [3] that S is always symmetric, that is, for any integer 0 ≤ s ≤ g (a,b)s ∈ S if and only if g (a,b)− s 6∈ S.It follows thatnS = g (a,b)+12.We may now prove Theorem 1.Proof of Theorem 1. Let ε> 0 be fixed. We distinguish two cases.Case 1. Suppose that a > (log(ab))1+ε. Let us take c = ab/(log(ab))1+ε. It is known [1] that ifk ∈ [0, . . . , g (a,b)] thennS [0,k] =⌊kb⌋∑i=0(⌊k − i ba⌋+1).In our case, we obtain thatnS [0,c] ≤⌊ ca⌋+⌊ cb⌋(⌊c −ba⌋+1)+1 ≤⌊ ca⌋+⌊ cb⌋(⌊ ca⌋+1)+1≤ ca+ cb+ c2ab+1 = bc +ac + c2 +abab< 2c2 + c2 + c2ab= 4c2ab= 4ab(log(ab))2+2εwhere the last inequality holds since c > b > a.Due to the symmetry of S, we havencS [g (a,b)− c, g (a,b)] = nS [0,c] <4ab(log(ab))2+2ε. (1)Let π(x) be the number of primes integers less or equals to x. We haveπ(g (a,b))−π(g (a,b)− c) >> clog(ab)= ab(log(ab))2+ε(2)when ab is large enough. The latter follows from Prime Number Theorem for short intervals(when c = ab/(log(ab))1+ε is large enough in comparison to g (a,b) = ab −a −b).Finally, by combining equations (1) and (2), we obtainπS ≥πS [g (a,b)− c, g (a,b)] ≥π(g (a,b))−π(g (a,b)− c)−ncS [g (a,b)− c, g (a,b)]> ab(log(ab))2+ε− 4ab(log(ab))2+2ε> 0where the last inequality holds since (log(ab))ε > 4 for ab large enough for the fixed ε. The aboveleads to the desired estimate of πS .C. R. Mathématique, 2020, 358, n 9-10, 1001-1004J.L. Ramírez Alfonsín and M. Skałba 1003Case 2. Suppose that 3 ≤ a ≤ (log(ab))1+ε.If p ∈ [b, . . . , g (a,b)] is a prime and p ≡ b (mod a) then p is clearly representable asp = b + p−ba a. By Siegel–Walfisz theorem [2, 7], the number of such primes p, denoted by N , isN = 1ϕ(a)∫ g (a,b)bdulogu+Rwhere ϕ is the Euler totient function and |R| < D ′(ε) g (a,b)(log(g (a,b)))2+2ε uniformly in a and g (a,b).Since the function 1/logu is decreasing on the interval [b, g (a,b)] then∫ g (a,b)bdulogu> (g (a,b)−b) · 1log g (a,b)and thereforeN > 1ϕ(a)· g (a, b)−blog(g (a, b))−D ′(ε) g (a, b)(log(g (a, b)))2+2ε. (3)Now, we have that1ϕ(a)· g (a,b)−blog(g (a,b))· (log(g (a,b)))2+εg (a,b)= 1ϕ(a)log(g (a,b))1+ε(1− bg (a,b))> 1log(ab)1+εlog(g (a,b))1+ε(1− bg (a,b))(since(log(ab))1+ε ≥ a >ϕ(a))>(log(ab)− log(3)log(ab))1+ε 15> F > 0(since g (a,b) > ab/3 and bg (a,b)≤ 45)for some absolute F > 0, uniformly for ab ≥ D ′′(ε) with a ≥ 3.It yields to1ϕ(a)· g (a,b)−blog(g (a,b))≥ F g (a,b)log(g (a,b))2+ε(4)and combining equations (3) and (4) we obtainN > F ′ g (a,b)log(g (a,b))2+εfor ab large enough for the fixed ε. The latter leads to the desired estimate of πS also in this case.1. Concluding remarksA number of computer experiments lead us to the following.Conjecture 2. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroupgenerated by a and b. Then,πS > 0.In analogy with the symmetry of 〈a,b〉mentioned above, our task of looking for primes in 〈a,b〉is related with the task of finding primes in [g (a,b)− 1)/2, . . . , g (a,b)]. From this point of view,Conjecture 2 can be thought of as a counterpart of the famous Chebyshev theorem stating thatthere is always a prime in [n, . . . , 2n] for any n ≥ 2, see [4, Chapter 3]. A way to attack Conjecture 2could be by applying effective versions of Siegel–Walfisz theorem. For instance, one may try touse [6, Corollary 8.31] in order to get computable constants in our estimates. However, it is notan easy task to trace all constants appearing in the relevant estimates of L(x,χ) (but in principlepossible). The remaining cases for small values ab must to be treated by computer.C. R. Mathématique, 2020, 358, n 9-10, 1001-10041004 J.L. Ramírez Alfonsín and M. SkałbaConjecture 3. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroupgenerated by a and b. Then,πS ∼ π(g (a,b))2for a →∞.In the same spirit as the prime number theorem, this conjecture seems to be out of reach.The famous Linnik’s theorem asserts that there exist absolute constants C and L such that: forgiven relatively prime integers a,b the least prime p satisfying p ≡ b (mod a) is less than C aL . Itis conjectured that one can take L = 2, but the current record is only that L ≤ 5 is allowed, see [8].On the same flavor of Linnik’s theorem that concerns the existence of primes of the form ax+b,Theorem 1 is concerning the existence of primes of the form ax+by with x, y ≥ 1 less than ab forsufficiently large ab. This relation could shed light on in either direction.References[1] G. Márquez-Campos, J. L. Ramírez Alfonsín, J. M. Tornero, “Integral points in rational polygons: a numerical semi-group approach”, Semigroup Forum 94 (2017), no. 1, p. 123-138.[2] K. Prachar, Primzahlverteilung, Die Grundlehren der Mathematischen Wissenschaften, vol. 91, Springer, 1957.[3] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications,vol. 30, Oxford University Press, 2005.[4] W. Sierpiński, Elementary Theory of Numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, PWN-PolishScientific Publisher; North-Holland, 1988.[5] J. J. Sylvester, “Question 7382”, Mathematical Questions from Educational Times 41 (1884), p. 21.[6] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd expanded ed., Graduate Studies inMathematics, vol. 163, American Mathematical Society, 2015.[7] A. Walfisz, “Zur additiven Zahlentheorie. II”, Math. Z. 40 (1936), no. 1, p. 592-607.[8] T. Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischenProgression, Bonner Mathematische Schriften, vol. 404, University Bonn; Mathematisches Institut and Mathematisch-Naturwissenschaftliche Fakultät, 2011.C. R. Mathématique, 2020, 358, n 9-10, 1001-1004

Primes in numerical semigroups

Comptes RendusMathématiqueJ.L. Ramírez Alfonsín and M. SkałbaPrimes in numerical semigroupsVolume 358, issue 9-10 (2020), p. 1001-1004.<https://doi.org/10.5802/crmath.104>© Académie des sciences, Paris and the authors, 2020.Some rights reserved.This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orgComptes RendusMathématique2020, 358, n 9-10, p. 1001-1004https://doi.org/10.5802/crmath.104Number Theory / Théorie des nombresPrimes in numerical semigroupsJ.L. Ramírez Alfonsín∗, a and M. Skałbaba UMI2924 - Jean- Christophe Yoccoz, CNRS-IMPA, Brazil and Univ. Montpellier,CNRS, Montpellier, Franceb Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa,PolandE-mails: jorge.ramirez-alfonsin@umontpellier.fr (J.L. Ramírez Alfonsín),skalba@mimuw.edu.pl (M. Skałba)Abstract. Let 0 < a < b be two relatively prime integers and let 〈a,b〉 be the numerical semigroup generatedby a and b with Frobenius number g (a,b) = ab−a−b. In this note, we prove that there exists a prime numberp ∈ 〈a,b〉 with p < g (a,b) when the product ab is sufficiently large. Two related conjectures are posed anddiscussed as well.2020 Mathematics Subject Classification. 11D07, 11N13.Funding. J.L. Ramírez Alfonsín was partially supported by Grant MATHAM-SUD 18-MATH-01, ProjectFLaNASAGraTA and INSMI-CNRS..Manuscript received 22nd July 2020, revised and accepted 30th July 2020.Let 0 < a < b be two relatively prime integers. Let S = 〈a,b〉 = {n |n = ax +by, x, y ∈Z, x, y ≥ 0}be the numerical semigroup generated by a and b. A well-known result due to Sylvester [5] statesthat the largest integer not belonging to S, denoted by g (a,b), is given by ab − a − b. g (a,b) iscalled the Frobenius number (we refer the reader to [3] for an extensive literature on the Frobeniusnumber).We clearly have that any prime p larger than g (a,b) belongs to 〈a,b〉. A less obvious and moreintriguing question is whether there is a prime p ≤ g (a,b) belonging to 〈a,b〉.In this note, we show that there always exists a prime p ∈ 〈a,b〉, p < g (a,b) when the productab is sufficiently large. The latter is a straight forward consequence of the below Theorem.Let 0 < u < v be integers. We defineπS [u, v] = |{p prime | p ∈ S, u ≤ p ≤ v}|.For short, we may write πS instead of πS [0, g (a,b)].Theorem 1. Let 3 ≤ a < b be two relatively prime integers and let S = 〈a,b〉 be the numericalsemigroup generated by a and b. Then, for any fixed ε> 0 there exists C (ε) > 0 such thatπS >C (ε) g (a,b)log(g (a,b))2+ε∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/1002 J.L. Ramírez Alfonsín and M. Skałbafor ab sufficiently large.Let us quickly introduce some notation and recall some facts needed for the proof of Theo-rem 1.Let S = 〈a,b〉 and let 0 < u < v be integers. We definenS [u, v] = |{n ∈N | u ≤ n ≤ v, n ∈ S}|andncS [u, v] = |{n ∈N | u ≤ n ≤ v, n 6∈ S}|.For short, we may write nS instead of nS [0, g (a,b)] and ncS instead of ncS [0, g (a,b)]. The set ofelements in ncS =N\ S are usually called the gaps of S.It is known [3] that S is always symmetric, that is, for any integer 0 ≤ s ≤ g (a,b)s ∈ S if and only if g (a,b)− s 6∈ S.It follows thatnS = g (a,b)+12.We may now prove Theorem 1.Proof of Theorem 1. Let ε> 0 be fixed. We distinguish two cases.Case 1. Suppose that a > (log(ab))1+ε. Let us take c = ab/(log(ab))1+ε. It is known [1] that ifk ∈ [0, . . . , g (a,b)] thennS [0,k] =⌊kb⌋∑i=0(⌊k − i ba⌋+1).In our case, we obtain thatnS [0,c] ≤⌊ ca⌋+⌊ cb⌋(⌊c −ba⌋+1)+1 ≤⌊ ca⌋+⌊ cb⌋(⌊ ca⌋+1)+1≤ ca+ cb+ c2ab+1 = bc +ac + c2 +abab< 2c2 + c2 + c2ab= 4c2ab= 4ab(log(ab))2+2εwhere the last inequality holds since c > b > a.Due to the symmetry of S, we havencS [g (a,b)− c, g (a,b)] = nS [0,c] <4ab(log(ab))2+2ε. (1)Let π(x) be the number of primes integers less or equals to x. We haveπ(g (a,b))−π(g (a,b)− c) >> clog(ab)= ab(log(ab))2+ε(2)when ab is large enough. The latter follows from Prime Number Theorem for short intervals(when c = ab/(log(ab))1+ε is large enough in comparison to g (a,b) = ab −a −b).Finally, by combining equations (1) and (2), we obtainπS ≥πS [g (a,b)− c, g (a,b)] ≥π(g (a,b))−π(g (a,b)− c)−ncS [g (a,b)− c, g (a,b)]> ab(log(ab))2+ε− 4ab(log(ab))2+2ε> 0where the last inequality holds since (log(ab))ε > 4 for ab large enough for the fixed ε. The aboveleads to the desired estimate of πS .C. R. Mathématique, 2020, 358, n 9-10, 1001-1004J.L. Ramírez Alfonsín and M. Skałba 1003Case 2. Suppose that 3 ≤ a ≤ (log(ab))1+ε.If p ∈ [b, . . . , g (a,b)] is a prime and p ≡ b (mod a) then p is clearly representable asp = b + p−ba a. By Siegel–Walfisz theorem [2, 7], the number of such primes p, denoted by N , isN = 1ϕ(a)∫ g (a,b)bdulogu+Rwhere ϕ is the Euler totient function and |R| < D ′(ε) g (a,b)(log(g (a,b)))2+2ε uniformly in a and g (a,b).Since the function 1/logu is decreasing on the interval [b, g (a,b)] then∫ g (a,b)bdulogu> (g (a,b)−b) · 1log g (a,b)and thereforeN > 1ϕ(a)· g (a, b)−blog(g (a, b))−D ′(ε) g (a, b)(log(g (a, b)))2+2ε. (3)Now, we have that1ϕ(a)· g (a,b)−blog(g (a,b))· (log(g (a,b)))2+εg (a,b)= 1ϕ(a)log(g (a,b))1+ε(1− bg (a,b))> 1log(ab)1+εlog(g (a,b))1+ε(1− bg (a,b))(since(log(ab))1+ε ≥ a >ϕ(a))>(log(ab)− log(3)log(ab))1+ε 15> F > 0(since g (a,b) > ab/3 and bg (a,b)≤ 45)for some absolute F > 0, uniformly for ab ≥ D ′′(ε) with a ≥ 3.It yields to1ϕ(a)· g (a,b)−blog(g (a,b))≥ F g (a,b)log(g (a,b))2+ε(4)and combining equations (3) and (4) we obtainN > F ′ g (a,b)log(g (a,b))2+εfor ab large enough for the fixed ε. The latter leads to the desired estimate of πS also in this case.1. Concluding remarksA number of computer experiments lead us to the following.Conjecture 2. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroupgenerated by a and b. Then,πS > 0.In analogy with the symmetry of 〈a,b〉mentioned above, our task of looking for primes in 〈a,b〉is related with the task of finding primes in [g (a,b)− 1)/2, . . . , g (a,b)]. From this point of view,Conjecture 2 can be thought of as a counterpart of the famous Chebyshev theorem stating thatthere is always a prime in [n, . . . , 2n] for any n ≥ 2, see [4, Chapter 3]. A way to attack Conjecture 2could be by applying effective versions of Siegel–Walfisz theorem. For instance, one may try touse [6, Corollary 8.31] in order to get computable constants in our estimates. However, it is notan easy task to trace all constants appearing in the relevant estimates of L(x,χ) (but in principlepossible). The remaining cases for small values ab must to be treated by computer.C. R. Mathématique, 2020, 358, n 9-10, 1001-10041004 J.L. Ramírez Alfonsín and M. SkałbaConjecture 3. Let 2 ≤ a < b be two relatively prime integers and let S be the numerical semigroupgenerated by a and b. Then,πS ∼ π(g (a,b))2for a →∞.In the same spirit as the prime number theorem, this conjecture seems to be out of reach.The famous Linnik’s theorem asserts that there exist absolute constants C and L such that: forgiven relatively prime integers a,b the least prime p satisfying p ≡ b (mod a) is less than C aL . Itis conjectured that one can take L = 2, but the current record is only that L ≤ 5 is allowed, see [8].On the same flavor of Linnik’s theorem that concerns the existence of primes of the form ax+b,Theorem 1 is concerning the existence of primes of the form ax+by with x, y ≥ 1 less than ab forsufficiently large ab. This relation could shed light on in either direction.References[1] G. Márquez-Campos, J. L. Ramírez Alfonsín, J. M. Tornero, “Integral points in rational polygons: a numerical semi-group approach”, Semigroup Forum 94 (2017), no. 1, p. 123-138.[2] K. Prachar, Primzahlverteilung, Die Grundlehren der Mathematischen Wissenschaften, vol. 91, Springer, 1957.[3] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series in Mathematics and its Applications,vol. 30, Oxford University Press, 2005.[4] W. Sierpiński, Elementary Theory of Numbers, 2nd ed., North-Holland Mathematical Library, vol. 31, PWN-PolishScientific Publisher; North-Holland, 1988.[5] J. J. Sylvester, “Question 7382”, Mathematical Questions from Educational Times 41 (1884), p. 21.[6] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd expanded ed., Graduate Studies inMathematics, vol. 163, American Mathematical Society, 2015.[7] A. Walfisz, “Zur additiven Zahlentheorie. II”, Math. Z. 40 (1936), no. 1, p. 592-607.[8] T. Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kleinste Primzahl in einer arithmetischenProgression, Bonner Mathematische Schriften, vol. 404, University Bonn; Mathematisches Institut and Mathematisch-Naturwissenschaftliche Fakultät, 2011.C. R. Mathématique, 2020, 358, n 9-10, 1001-1004

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