Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are saller tha +, ad ote that oe of the ubers N +, N + 3, N + 4,...,N +, N + + is prie, sice for i + we ow that i has a prie factor that is saller tha +, ad this factor also divides N, ad hece also N + i. With this recipe, we fid, for exaple, for = 0 that oe of the te ubers 3, 33, 34,..., 3 is prie. But there are also upper bouds for the gaps i the sequece of prie ubers. A faous boud states that the gap to the ext prie caot be larger tha the uber we start our search at. This is ow as Bertrad s postulate, sice it was cojectured ad verified epirically for < 3 000 000 by Joseph Bertrad. It was first proved for all by Pafuty Chebyshev i 850. A uch sipler proof was give by the Idia geius Raauja. Our Boo Proof is by Paul Erdős: it is tae fro Erdős first published paper, which appeared i 93, whe Erdős was 9. Joseph Bertrad Bertrad s postulate. For every, there is soe prie uber p with < p. Proof. We will estiate the size of the bioial coefficiet carefully eough to see that if it did t have ay prie factors i the rage < p, the it would be too sall. Our arguet is i five steps. We first prove Bertrad s postulate for < 4000. For this oe does ot eed to chec 4000 cases: it suffices this is Ladau s tric to chec that, 3, 5, 7, 3, 3, 43, 83,63, 37, 63, 59, 503, 400 is a sequece of prie ubers, where each is saller tha twice the previous oe. Hece every iterval {y : < y }, with 4000, cotais oe of these 4 pries.
8 Bertrad s postulate Next we prove that p 4 x for all real x, p x where our otatio here ad i the followig is eat to iply that the product is tae over all prie ubers p x. The proof that we preset for this fact uses iductio o the uber of these pries. It is ot fro Erdős origial paper, but it is also due to Erdős see the argi, ad it is a true Boo Proof. First we ote that if q is the largest prie with q x, the p ad 4 q 4 x. p = p x p q Thus it suffices to chec for the case where x = q is a prie uber. For q = we get 4, so we proceed to cosider odd pries q = +. Here we ay assue, by iductio, that is valid for all itegers x i the set {, 3,..., }. For q = + we split the product ad copute p = + p p 4 4 = 4. p + p + +<p + All the pieces of this oe-lie coputatio are easy to see. I fact, p 4 p + holds by iductio. The iequality +<p + p + Legedre s theore The uber! cotais the prie factor p exactly X j p ties. Proof. Exactly p of the factors of! = 3 are divisible by p, which accouts for p p-factors. Next, p of the factors of! are eve divisible by p, which accouts for the ext p prie factors p of!, etc. follows fro the observatio that + = +!!+! is a iteger, where the pries that we cosider all are factors of the uerator +!, but ot of the deoiator! +!. Fially + holds sice + ad + + are two equal! suads that appear i + + = +. =0!! co- 3 Fro Legedre s theore see the box we get that =! tais the prie factor p exactly p p
Bertrad s postulate 9 ties. Here each suad is at ost, sice it satisfies p p < p p =, ad it is a iteger. Furtherore the suads vaish wheever p >. Thus cotais p exactly p p ax{r : p r } ties. Hece the largest power of p that divides is ot larger tha. I particular, pries p > appear at ost oce i. Furtherore ad this, accordig to Erdős, is the ey fact for his proof pries p that satisfy 3 < p do ot divide Exaples such as at all! Ideed, `6 3p > iplies for 3, ad hece p 3 that p ad p are the oly ultiples of p that appear as factors i the uerator of!!!, while we get two p-factors i the deoiator. 4 Now we are ready to estiate. For 3, usig a estiate fro page for the lower boud, we get 4 p <p 3 p <p ad thus, sice there are ot ore tha pries p, 4 + p p for 3. <p 3 <p 5 Assue ow that there is o prie p with < p, so the secod product i is. Substitutig ito we get or 4 + 4 3 4 3 +, 3 which is false for large eough! I fact, usig a + < a which holds for all a, by iductio we get = 6 6 6 6 < + < 6 6 6 6, 4 ad thus for 50 ad hece 8 < we obtai fro 3 ad 4 3 + < 6 8+8 < 0 6 = 0/3. p 3 = 3 5 7 7 9 3 `8 4 = 3 3 3 5 7 9 3 `30 5 = 4 3 5 7 9 3 9 illustrate that very sall prie factors p < ca appear as higher powers i `, sall pries with < 3 p appear at ost oce, while factors i the gap with < p 3 do t appear at all. This iplies /3 < 0, ad thus < 4000.
0 Bertrad s postulate Oe ca extract eve ore fro this type of estiates: Fro oe ca derive with the sae ethods that p 30 for 4000, <p ad thus that there are at least log 30 = 30 log + pries i the rage betwee ad. This is ot that bad a estiate: the true uber of pries i this rage is roughly / log. This follows fro the prie uber theore, which says that the liit #{p : p is prie} li / log exists, ad equals. This faous result was first proved by Hadaard ad de la Vallée-Poussi i 896; Selberg ad Erdős foud a eleetary proof without coplex aalysis tools, but still log ad ivolved i 948. O the prie uber theore itself the fial word, it sees, is still ot i: for exaple a proof of the Riea hypothesis see page 49, oe of the ajor usolved ope probles i atheatics, would also give a substatial iproveet for the estiates of the prie uber theore. But also for Bertrad s postulate, oe could expect draatic iproveets. I fact, the followig is a faous usolved proble: Is there always a prie betwee ad +? For additioal iforatio see [3, p. 9] ad [4, pp. 48, 57]. Appedix: Soe estiates Estiatig via itegrals There is a very siple-but-effective ethod of estiatig sus by itegrals as already ecoutered o page 4. For estiatig the haroic ubers H = = = we draw the figure i the argi ad derive fro it H = < dt = log t by coparig the area below the graph of ft = t t with the area of the dar shaded rectagles, ad H = > = dt = log t
Bertrad s postulate by coparig with the area of the large rectagles icludig the lightly shaded parts. Tae together, this yields log + < H < log +. I particular, li H, ad the order of growth of H is give by H li log =. But uch better estiates are ow see [], such as Here O ` deotes a fuctio f 6 H such that f c holds for soe = log + γ + + 0 4 + O 6, 6 costat c. where γ 0.577 is Euler s costat. Estiatig factorials Stirlig s forula The sae ethod applied to yields log! = log + log 3 +... + log = log! < where the itegral is easily coputed: [ log t dt = t log t t log = log t dt < log!, ] = log +. Thus we get a lower estiate o!! > e log + = e e ad at the sae tie a upper estiate.! =! < e log + = e e Here a ore careful aalysis is eeded to get the asyptotics of!, as give by Stirlig s forula! π. e Ad agai there are ore precise versios available, such as! = π + e + 88 39 540 3 + O 4. Here f g eas that f li g =. Estiatig bioial coefficiets Just fro the defiitio of the bioial coefficiets as the uber of -subsets of a -set, we ow that the sequece 0,,..., of bioial coefficiets
Bertrad s postulate 3 3 4 5 6 5 06 4 0 0 5 5 6 7 35 35 7 Pascal s triagle sus to =0 = is syetric: =. Fro the fuctioal equatio = + oe easily fids that for every the bioial coefficiets for a sequece that is syetric ad uiodal: it icreases towards the iddle, so that the iddle bioial coefficiets are the largest oes i the sequece: = 0 < < < / = / > > > =. Here x resp. x deotes the uber x rouded dow resp. rouded up to the earest iteger. Fro the asyptotic forulas for the factorials etioed above oe ca obtai very precise estiates for the sizes of bioial coefficiets. However, we will oly eed very wea ad siple estiates i this boo, such as the followig: for all, while for we have /, with equality oly for =. I particular, for, 4. This holds sice /, a iddle bioial coefficiet, is the largest etry i the sequece 0 +,,,...,, whose su is, ad whose average is thus. O the other had, we ote the upper boud for bioial coefficiets + =!!, which is a reasoably good estiate for the sall bioial coefficiets at the tails of the sequece, whe is large copared to. Refereces [] P. ERDŐS: Beweis eies Satzes vo Tschebyschef, Acta Sci. Math. Szeged 5 930-3, 94-98. [] R. L. GRAHAM, D. E. KNUTH & O. PATASHNIK: Cocrete Matheatics. A Foudatio for Coputer Sciece, Addiso-Wesley, Readig MA 989. [3] G. H. HARDY & E. M. WRIGHT: A Itroductio to the Theory of Nubers, fifth editio, Oxford Uiversity Press 979. [4] P. RIBENBOIM: The New Boo of Prie Nuber Records, Spriger-Verlag, New Yor 989.
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