Digitally delicate primes

Transcription

1 Digitally elicate rimes Jackson Hoer Paul Pollack Deartment of Mathematics University of Georgia Athens, Georgia Tao has shown that in any fixe base, a ositive roortion of rime numbers cannot have any igit change an remain rime. In other wors, most rimes are igitally elicate. We strengthen this result in a manner suggeste by Tao: A ositive roortion of rimes become comosite uner any change of a single igit an any insertion a fixe number of arbitrary igits at the beginning or en. Introuction In a short note ublishe in 2008, Tao [] rove the following theorem: Theorem.. Let K 2 be an integer. For all sufficiently large integers, the number of rimes between an + /K) such that k + ja i is comosite for all integers a, j, k K an 0 i K log is at least c K log for some constant c K > 0 eening on only K. The following consequence is immeiate, in view of the rime number theorem or Chebyshev s weaker estimates). Corollary.2. Fix a base a 2. A ositive roortion of rime numbers become comosite if any single igit in their base a exansion is altere. The infinitue of the rimes aearing in Corollary.2 ha earlier been shown by Erős [2]. He assumes a = 0 but the argument generalizes in an obvious way.) When a = 0, these igitally elicate rimes are tabulate as sequence A in the OEIS, where they are calle weakly rime. At the conclusion of [], Tao suggests a few ways his result coul ossibly be imrove. In this aer we establish one of the suggeste generalizations: Theorem.3. Fix an integer K 2. There is a constant c K > 0 such that the following hols for all sufficiently large : Let S [ K, K] be an arbitrary set of integers of carinality at most K. Let K be the number of rimes + /K) such that k + ja i + s is either equal to or comosite for all combinations of integers a, i, j, k, an s where a, j, k K, 0 i K log, an s S. Then K c K This immeiately yiels the following strengthening of Corollary.2. log. jacksonh@uga.eu ollack@uga.eu

2 Corollary.4. In any fixe base, a ositive roortion of rime numbers become comosite if one moifies any single igit an aens a boune number of igits at the beginning or en. As in Tao s work, the key iea of the roof is to use a artial covering along with an uer boun sieve. The following well-known estimate lays a critical role see [5, Theorem 2.2, 68], [, Corollary A.2]). Lemma.5 Brun/Selberg uer boun). Let W an b be ositive integers an let k an h be non-zero integers. If x is sufficiently large eening on W an b), the number of rimes m x where m b mo W ) an km + h is also rime is k x W log x) 2 W where the roucts are restricte to rime numbers. ) ) 2 ) ), h W Whenever Lemma.5 is alie in Tao s roof of Theorem., the rouct over iviing h is uniformly boune. However, to rove Theorem.3, we must eal with cases where that rouct can be very large. To work aroun this, we show that such cases arise very rarely, so rarely that this rouct is boune in a suitable average sense. To establish this, we nee to invoke a classical theorem of Romanoff [8] about multilicative orers, which originally aeare in his work on numbers of the form + 2 k. Actually we use a slightly strengthene form of Romanoff s result ue to Erős [].) We woul like to raw the intereste reaer s attention to the work reorte on in [3], [7], an [4], which also concerns roblems connecte with rimality an igital exansions. otation We write n to inicate that n is squarefree. The letter always enotes a rime. For a given integer n we use P n) to enote the largest rime ivisor of n an ωn) for the number of istinct rime factors of n. For a given integer a, we write l a ) for the multilicative orer of a moulo. This notation reflects the imortance in our analysis of consiering l a ) rimarily as a function of rather than as a function of a. We use f = Og), or f g, to mean that f Cg for a suitable constant C. We use f g synonymously with g f. If f g f, we write f g. We use f = og) to mean lim f/g = 0 as, holing other variables constant. In what follows, imlie constants may een on K. Any further eenence or ineenence) will be secifie exlicitly. 2 Proof of Theorem.3 2. A selective search We will confine our search for igitally elicate rimes to rimes lying in a certain conveniently chosen invertible resiue class b mo W. Here b mo W lays the same role for us as in Tao s aer []: It is selecte so that whenever b mo W ) is rime, k + ja i + s has a known rime factor for the vast majority of choices of a, i, j, k, an s as mae recise in 3) below). For the remaining choices of a, i, j, k, an s, the uer boun sieve rovies sufficient control on the number of with k + ja i + s rime. To secify the resiue class b mo W, we will require the use of a hanful rimes, etermine by K an an integer M K. 2

3 Lemma 2.. Let K 2 be an integer an let M K also be an integer. There is a set P that is the isjoint union of sets P = K a=2 P a, where for each 2 a K, P a is a finite set of rimes such that: i) for all P a, we have q := P a ) > K, ii) the rimes q are istinct for istinct P, iii) P a M. Proof. Accoring to a theorem of Stewart [9, Theorem ], P a ) log for all rimes an all 2 a K. See [0] for a more recent, much stronger estimate.) Keeing this min, we construct the sets P a inuctively. Given an integer a with 2 a K, assume that the sets P n have been constructe for all integers 2 n < a. We construct P a as follows. By Stewart s result, we can ick 0 so that whenever > 0, we have P a ) larger than K an larger than any element of q for 2 n<a P n. As runs through the consecutive rimes succeeing 0, the numbers P a ) are istinct, since the orer of a moulo P a ) is recisely. So we can construct P a as the set of the first several consecutive rimes exceeing 0. Here first several means that we continue aing rimes to P a until iii) hols. This is ossible ue to the ivergence of / when is taken over all rimes. We now set W = P q. Observe that from Stewart s theorem quote above, W = q P = O); ) log here the final estimate follows, for examle, by artial summation along with the rime number theorem. Assume M is sufficiently large in terms of K. Then we can artition each P a into isjoint sets P a,j,k,s such that P a = K an for each P a,j,k,s we have P a,j,k,s j K k= s S M. 2) P a,j,k,s Recall that by our convention, imlie constants may een on K.) We now make our choice of resiue class b mo W. Suose 2 a K, j, k K, an s S. Let P a,j,k,s. Since q > K k, we know that k exists moulo q. Moreover, at least one of the two resiue classes k j + s) mo q or k ja + s) mo q is invertible. Pick one, an say it is b mo W. We etermine b mo W as the solution to the simultaneous congruences b b mo q ) for all P. ote that b mo W is inee a corime resiue class. 3

4 2.2 Some initial reuctions In what follows, we will always assume is sufficiently large in terms of fixe arameters M an K. Ultimately, M will be chosen sufficiently large in terms of K.) Let Q := #{m [, + K )] : m b mo W ), m rime}. By the rime number theorem for arithmetic rogressions, Q φw ) log. We woul like to show that the same lower boun hols even after removing from our count those m having km + ja i + s noncomosite an ) for some a, j, k K, 0 i K log, an s S. We first isense with those cases when km + ja i + s is noncomosite in virtue of having km + ja i + s. Let E := #{m [, + K )] : m b mo W ), m rime, km + ja i + s for some value of a, i, j, k, s} A given combination of a, i, j, k, an s can contribute only O) elements m to E, so we have E log. This boun is clearly oq ), an so is negligible for us. It remains to iscar those m having km + ja i + s rime an m) for some a, i, j, k, s as above. We may assume ja i + s 0. Otherwise km is rime, forcing k = an km + ja i + s = m, contrary to hyothesis. The next easiest series of cases correson to a =. In these cases, km + j + s is rime an m) for some j, k K an s S. Given j, k, an s, the number of m we must iscar here is, by Lemma.5, W log ) 2 ) 2 ). W j+s From ), the rouct on iviing W is O). Since 0 < j + s K + ), the rouct on iviing j + s cannot excee Olog log ); see [6, Theorem 328,. 352]. Summing on the O) ossibilities for j, k, s, we see we must iscar a total of W log log log ) 2 rimes m from these cases. This is oq ). aturally, the heart of the roof is the consieration of those cases when a 2. Let Q,a,i,j,k,s := #{m [, + )] : m b mo W ), K m rime, an km + ja i + s rime an m}. In the next section, we will show that K K a=2 j K k= s S Q,a,i,j,k,s W log ex 2 θ KM) 3) for a certain constant θ K > 0. Fixing M sufficiently large in terms of K, we see that these values of a force us to iscar at most say) 2 Q rimes. Collecting the above estimates, we fin that there are Q / log remaining rimes m, all of which are igitally elicate in the strong sense of Theorem.3. 4

5 2.3 Detaile counting In this section, we establish the claime uer boun on K K a=2 j K k= s S Q,a,i,j,k,s. We first hanle the sum on i. For now treat a, j, k, an s as fixe an consier Q,a,i,j,k,s. 4) Because of our careful choice of b, either kb + j + s 0 mo q ), or kb + ja + s 0 mo q ) for all our P a,j,k,s. In the former case, if i 0 mo ) for some P a,j,k,s, then q kb + ja i + s. In the latter case, the same ivisibility hols instea when i mo ). To see these results, recall that a mo q ), by the choice of q.) If q kb + ja i + s then at most two values of m for a given a, i, j, k, an s can have km + ja i + s rime: those where km + ja i + s = q. So the number of m contribute to 4) in this way is Olog ). We thus focus on the remaining values of i. Let I := {0 i K log : for all P a,j,k,s, q kb + ja i + s}, where I is unerstoo to een on the given a, j, k, an s. The conition that q kb+ja i +s sieves out either those i 0 mo ) or those i mo ). Hence, the Chinese remainer theorem along with inclusion-exclusion yiels Moreover, #I P a,j,k,s Q,a,i,j,k,s log + i I ) log. 5) Q,a,i,j,k,s. 6) Whenever ja i + s = 0, the quantity Q,a,i,j,k,s vanishes, an so the final sum on i can be restricte to those values with ja i + s 0. By another alication of Lemma.5, as long as ja i + s 0, Q,a,i,j,k,s W log ) 2 ja i +s ). 7) We omitte the rouct over iviing W here, since ) shows that rouct is.) Controlling the contribution from the rouct terms in 7) requires some care, an this is the main novelty of the aer. The corresoning rouct in Tao s work [] is only over iviing ja i, an so is trivially O).) To this en, we aly the Cauchy Schwarz inequality to euce that i I ja i +s ) i I ) /2 ja i +s ) 2 ) /2. 8) The first right-han sum simly counts the number of i I, an so from 2) an 5), = #I ) log ex θ K M) log, 9) i I P a,j,k,s 5

6 for a constant θ K > 0. To estimate the secon sum of 8), we begin by observing that ) 2 = ) + ), an that ) ) ex <, where the roucts an sums are over all rimes. Thus, ja i +s ) 2. ja i +s We claim that truncating the last rouct to rimes log will not change its magnitue. To see this, observe that ja i +s >log 2 ex log ja i +s >log ) 2 ex log log ja i ) + s. log log Put Z := K K K log +K. Since ja i +s j a i + s Z, we have log ja i + s log Z log, an so final exression in the receing islay is O). Consequently,, ja i +s ja i +s log as claime. ow rewrite ja i +s log = ja i +s log 2 ω). Assembling the above, we can estimate the secon sum in 8) as follows: ja i +s ) 2 Z log ja i +s log 2 ω) 2 ω) ja i +s Suose i is such that ja i + s. Defining B := gc, ja), an keeing in min that is squarefree, we fin that gc, s) = gc, ja i ) = gc, ja) = B, 0). an ja B a i s B mo B ). 6

7 This congruence, along with 0), shows that a i belongs to a uniquely etermine corime resiue class moulo /B. Thus, i belongs to a fixe resiue class moulo l a /B ), an so 2 ω) 2 ω) K log ) + Z Z l a B log ja i s mo ) log log /B,a)= 2 ω) l a B ) + To hanle the first right-han sum, write = B. Since B ja, a,/b )= 2 ω) l a B ) B ja a, )= 2 ωb ) B l a ) B ja Z log 2 ωb) B a, )= 2 ω). ) 2 ω ) l a ), ue to the comlete subaitivity of ω. Erős has roven a strengthening of Romanoff s theorem [, see Lemma 2,. 47] saying that for any two ositive integers A an S, the series n n,a)= S ωn) n l A n) is convergent. Taking n =, A = a, an S = 2, an noting that a, j, an B are all O), we see that the first of the two summans in ) is Olog ). To eal with the secon summan of ), we reverse a revious ste an rewrite Z log 2 ω) = log log ) By Mertens Theorem see [6, Theorem 429,. 466]), the final exression is Olog log ) 2 ), which is certainly Olog ). Hence, ) 2 log. ja i +s Substituting this estimate an 9) into 8), ) log ex ) 2 θ KM. i I ja i +s We now euce from 6) that Q,a,i,j,k,s W log ex ) 2 θ KM. Finally, summing over the O) ossibilities for a, j, k, an s yiels 3) an so comletes the roof. 2. 7

8 Acknowlegments We woul like to thank UGA s Center for Unergrauate Research Oortunities CURO) for the oortunity to work together. Work of the first author is suorte by the 205 CURO Summer Fellowshi, an work of the secon author is suorte by SF awar DMS We are grateful to Christian Elsholtz for insightful comments an we thank the referee for a careful reaing. References [] P. Erős, On some roblems of Bellman an a theorem of Romanoff, J. Chinese Math. Soc..S.) 95), [2], Solution to roblem 029: Erős an the comuter, Mathematics Magazine ), [3] M. Filaseta, M. Kozek, C. icol, an J.L. Selfrige, Comosites that remain comosite after changing a igit, J. Comb. umber Theory 2 200), ). [4] J. Grantham, W. Jarnicki, J. Rickert, an S. Wagon, Reeately aening any igit to generate comosite numbers, Amer. Math. Monthly 2 204), [5] H. Halberstam an H.-E. Richert, Sieve methos, Lonon Mathematical Society Monograhs, vol. 4, Acaemic Press, 974. [6] G.H. Hary an E.M. Wright, An introuction to the theory of numbers, sixth e., Oxfor University Press, Oxfor, [7] S. Konyagin, umbers that become comosite after changing one or two igits, resentation at Erős centennial conference, 203. Online at htt:// conferences/eros00/slies/konyagin.f. [8].P. Romanoff, Über einige Sätze er aitiven Zahlentheorie, Math. Ann ), [9] C.L. Stewart, On ivisors of Fermat, Fibonacci, Lucas, an Lehmer numbers, Proc. Lonon Math. Soc. 3) ), [0], On ivisors of Lucas an Lehmer numbers, Acta Math ), [] T. Tao, A remark on rimality testing an ecimal exansions, J. Aust. Math. Soc. 9 20),