aa r X i v : . [ m a t h . N T ] A p r PRIMES IN NUMERICAL SEMIGROUPS
J.L. RAMÍREZ ALFONSÍN AND M. SKAŁBA
Abstract.
Let < a < b be two relatively prime integers andlet h a, b i be the numerical semigroup generated by a and b with Frobenius number g ( a, b ) = ab − a − b . In this note, we prove thatthere exists a prime number p ∈ h a, b i with p < g ( a, b ) when theproduct ab is sufficiently large. Two related conjectures are posedand discussed as well. Let < a < b be two relatively prime integers. Let S = h a, b i = { n | n = ax + by, x, y ∈ Z , x, y ≥ } be the numerical semigroupgenerated by a and b . A well-known result due to Sylvester [6] statesthat the largest integer not belonging to S , denoted by g ( a, b ) , is givenby ab − a − b . g ( a, b ) is called the Frobenius number . We refer thereader to [3] for a complexity result on the Frobenius number for generalnumerical semigroups and to [4] for an extensive literature on it.We clearly have that any prime p larger than g ( a, b ) belongs to h a, b i .A less obvious and more intriguing question is whether there is a prime p ≤ g ( a, b ) belonging to h a, b i .In this note, we show that there always exists a prime p ∈ h a, b i , p Let ≤ a < b be two relatively prime integers and let S = h a, b i be the numerical semigroup generated by a and b . Then, forany fixed ε > there exists C ( ε ) > such that π S > C ( ε ) g ( a, b )log( g ( a, b )) ε for ab sufficiently large. Mathematics Subject Classification. Primary 11D07, 11N13, 11A41. Key words and phrases. Primes, numerical semigroups, Frobenius number. Partially supported by Program MATH AmSud, Grant MATHAMSUD 18-MATH-01, Project FLaNASAGraTA.. AND M. SKAŁBA Let us quickly introduce some notation and recall some facts neededfor the proof of Theorem 1.Let S = h a, b i and let < u < v be integers. We define n S [ u, v ] = |{ n ∈ N | u ≤ n ≤ v, n ∈ S }| and n cS [ u, v ] = |{ n ∈ N | u ≤ n ≤ v, n S }| . For short, we may write n S instead of n S [0 , g ( a, b )] and n cS insteadof n cS [0 , g ( a, b )] . The set of elements in n cS = N \ S are usually calledthe gaps of S .It is known [4] that S is always symmetric , that is, for any integer ≤ s ≤ g ( a, b ) s ∈ S if and only if g ( a, b ) − s S. Obtaining that n S = g ( a, b ) + 12 . Let π ( x ) be the number of primes integers less or equals to x for anyreal number x . We recall that the well-known prime number theoremasserts that(1) π ( x ) ∼ x log x . We may now prove Theorem 1. Proof of Theorem 1. Let ε > be fixed. We distinguish two cases. Case 1) Suppose that a > (log( ab )) ε . Let us take c = ab/ (log( ab )) ε .It is known [1] that if k ∈ [0 , . . . , g ( a, b )] then n S [0 , k ] = ⌊ kb ⌋ X i =0 (cid:18)(cid:22) k − iba (cid:23) + 1 (cid:19) . In our case, we obtain that n S [0 , c ] < ⌊ ca ⌋ + ⌊ cb ⌋ (cid:0) ⌊ c − ba ⌋ + 1 (cid:1) < ⌊ ca ⌋ + ⌊ cb ⌋ (cid:0) ⌊ ca ⌋ + 1 (cid:1) < ca + cb + c ab < bc + ac + c ab < c + c ab = c ab = ab (log( ab )) ε where the last inequality holds since c > b > a .Due to the symmetry of S , we have(2) n cS [ g ( a, b ) − c, g ( a, b )] = n S [0 , c ] < ab (log( ab )) ε . RIMES IN NUMERICAL SEMIGROUPS 3 Now, by (1), we have(3) π S [ g ( a, b ) − c, g ( a, b )] = π ( g ( a, b )) − π ( g ( a, b ) − c ) >> c log( ab ) = ab (log( ab )) ε when ab is large enough.Finally, by combining equations (2) and (3), we obtain π S > π S [ g ( a, b ) − c, g ( a, b )] − n cS [ g ( a, b ) − c, g ( a, b )] > ab (log( ab )) ε − ab (log( ab )) ε > where the last inequality holds since (log( ab )) ε > for ab largeenough for the fixed ǫ . The above leads to the desired estimation of π S . Case 2) Suppose that ≤ a ≤ (log( ab )) ε .If p ∈ [ b, . . . , g ( a, b )] is a prime and p ≡ b (mod a ) then p is clearlyrepresentable as p = b + p − ba a . By Siegel-Walfisz theorem [2, 8], thenumber of such primes p , denoted by N , is N = 1 ϕ ( a ) Z g ( a,b ) b du log u + R where ϕ is the Euler totient function and | R | < D ′ ( ε ) g ( a,b )(log( g ( a,b ))) ε uniformly in [ a, . . . , g ( a, b )] .Since the function / log u is decreasing on the interval [ b, g ( a, b )] then Z g ( a,b ) b du log u > ( g ( a, b ) − b ) · g ( a, b ) and therefore(4) N > ϕ ( a ) · g ( a, b ) − b log( g ( a, b )) − D ′ ( ε ) g ( a, b )(log( g ( a, b ))) ε . Now, we have that ϕ ( a ) · g ( a,b ) − b log( g ( a,b )) · (log( g ( a,b ))) ε g ( a,b ) = ϕ ( a ) log( g ( a, b )) ε (cid:16) − bg ( a,b ) (cid:17) > ab ) ε log( g ( a, b )) ε (cid:16) − bg ( a,b ) (cid:17) (since (log( ab )) ε ≥ a > ϕ ( a ) ) > (cid:16) log( ab ) − log(3)log( ab ) (cid:17) ε > F > (since g ( a, b )) > ab/ and bg ( a,b ) ≤ )for some absolute F > , uniformly for ab ≥ D ′′ ( ε ) with a ≥ . J.L. RAMÍREZ ALFONSÍN AND M. SKAŁBA Yielding to(5) ϕ ( a ) · g ( a, b ) − b log( g ( a, b )) ≥ F g ( a, b )log( g ( a, b )) ε and combining equations (4) and (5) we obtain N > F ′ g ( a, b )log( g ( a, b )) ε for ab large enough for the fixed ǫ . The latter leads to the desiredestimation of π S also in this case. (cid:4) Concluding remarks A number of computer experiments lead us to the following. Conjecture 1. Let ≤ a < b be two relatively prime integers and let S be the numerical semigroup generated by a and b . Then, π S > . In analogy with the symmetry of h a, b i mentioned above, our task oflooking for primes in h a, b i is related with the task of finding primesin [ g ( a, b ) − / , . . . , g ( a, b )] . From this point of view, Conjecture 1can be thought of as a counterpart of the famous Chebyshev theoremstating that there is always a prime in [ n, . . . , , n ] for any n ≥ , see [5,Chapter 3]. A way to attack Conjecture 1 could be by applying effectiveversions of Siegel-Walfisz theorem. For instance, one may try to use [7,Corollary 8.31] in order to get computable constants in our estimates.However, it is not an easy task to trace all constants appearing in therelevant estimates of L ( x, χ ) (but in principle possible). The remainingcases for small values ab must to be treated by computer. Conjecture 2. Let ≤ a < b be two relatively prime integers and let S be the numerical semigroup generated by a and b . Then, π S ∼ π ( g ( a, b ))2 for ab → ∞ . In the same spirit as the prime number theorem, this conjectureseems to be out of reach.The famous Linnik’s theorem asserts that there exist absolute con-stants C and L such that: for given relatively prime integers a, b theleast prime p satisfying p ≡ b (mod a ) is less than Ca L . It is conjec-tured that the value L = 2 , but the current record is only that L ≤ ,see [9]. RIMES IN NUMERICAL SEMIGROUPS 5 On the same flavor of Linnik’s theorem that concerns the existenceof primes of the form ax + b , Theorem 1 is concerning the existence ofprimes of the form ax + by with x, y ≥ less than ab for sufficientlylarge ab . This relation could shed light on in either direction. References [1] G. Márquez-Campos, J.L. Ramírez Alfonsín, J.M. Tornero, Integral points inrational polygons: a numerical semigroup approach, Semigroup Forum (1)(2017), 123-138.[2] K. Prachar, Primzahlverteilung, Die Grundlehren der Mathematischen Wis-senschaften XCI, Springer-Verlag 1957.[3] J.L. Ramírez Alfonsín, Complexity of the Frobenius problem, Combinatorica (1) (1996), 143-147.[4] J.L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford LectureSer. in Math. and its Appl. 30, Oxford University Press 2005.[5] W. Sierpinski, Elementary Theory of Numbers, Second Ed., PWN-Polish Sci-entific Publisher and North-Holland 1988,[6] J.J. Sylvester Question 7382 , Mathematical Questions from Educational Times41, 1884.[7] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory,Third Edition, Graduate Studies in Mathematics 163, AMS 2015.[8] A. Walfisz, Zur additiven Zahlentheorie. II Mathematische Zeitschrift (1)(1936) 592-607.[9] T. Xylouris, Über die Nullstellen der Dirichletschen L-Funktionen und die kle-inste Primzahl in einer arithmetischen Progression , Dissertation, Bonn 2011. Institute of Mathematics, University of Warsaw, Banacha 2, 02-097Warszawa, Poland E-mail address : [email protected] IMAG, Univ. Montpellier, CNRS, Montpellier, France and UMI2924- Jean-Christophe Yoccoz, CNRS-IMPA E-mail address ::