Tamar Ziegler and I have just uploaded to the arXiv our paper “Infinite partial sumsets in the primes“. This is a short paper inspired by a recent result of Kra, Moreira, Richter, and Robertson (discussed for instance in this Quanta article from last December) showing that for any set of natural numbers of positive upper density, there exists a sequence of natural numbers and a shift such that for all this answers a question of Erdős). In view of the “transference principle“, it is then plausible to ask whether the same result holds if is replaced by the primes. We can show the following results:

Theorem 1

- (i) If the Hardy-Littlewood prime tuples conjecture (or the weaker conjecture of Dickson) is true, then there exists an increasing sequence of primes such that is prime for all .
- (ii) Unconditionally, there exist increasing sequences and of natural numbers such that is prime for all .
- (iii) These conclusions fail if “prime” is replaced by “positive (relative) density subset of the primes” (even if the density is equal to 1).

We remark that it was shown by Balog that there (unconditionally) exist arbitrarily long but *finite* sequences of primes such that is prime for all . (This result can also be recovered from the later results of Ben Green, myself, and Tamar Ziegler.) Also, it had previously been shown by Granville that on the Hardy-Littlewood prime tuples conjecture, there existed increasing sequences and of natural numbers such that is prime for all .

The conclusion of (i) is stronger than that of (ii) (which is of course consistent with the former being conditional and the latter unconditional). The conclusion (ii) also implies the well-known theorem of Maynard that for any given , there exist infinitely many -tuples of primes of bounded diameter, and indeed our proof of (ii) uses the same “Maynard sieve” that powers the proof of that theorem (though we use a formulation of that sieve closer to that in this blog post of mine). Indeed, the failure of (iii) basically arises from the failure of Maynard’s theorem for dense subsets of primes, simply by removing those clusters of primes that are unusually closely spaced.

Our proof of (i) was initially inspired by the topological dynamics methods used by Kra, Moreira, Richter, and Robertson, but we managed to condense it to a purely elementary argument (taking up only half a page) that makes no reference to topological dynamics and builds up the sequence recursively by repeated application of the prime tuples conjecture.

The proof of (ii) takes up the majority of the paper. It is easiest to phrase the argument in terms of “prime-producing tuples” – tuples for which there are infinitely many with all prime. Maynard’s theorem is equivalent to the existence of arbitrarily long prime-producing tuples; our theorem is equivalent to the stronger assertion that there exist an infinite sequence such that every initial segment is prime-producing. The main new tool for achieving this is the following cute measure-theoretic lemma of Bergelson:

Lemma 2 (Bergelson intersectivity lemma)Let be subsets of a probability space of measure uniformly bounded away from zero, thus . Then there exists a subsequence such that for all .

This lemma has a short proof, though not an entirely obvious one. Firstly, by deleting a null set from , one can assume that all finite intersections are either positive measure or empty. Secondly, a routine application of Fatou’s lemma shows that the maximal function has a positive integral, hence must be positive at some point . Thus there is a subsequence whose finite intersections all contain , thus have positive measure as desired by the previous reduction.

It turns out that one cannot quite combine the standard Maynard sieve with the intersectivity lemma because the events that show up (which roughly correspond to the event that is prime for some random number (with a well-chosen probability distribution) and some shift ) have their probability going to zero, rather than being uniformly bounded from below. To get around this, we borrow an idea from a paper of Banks, Freiberg, and Maynard, and group the shifts into various clusters , chosen in such a way that the probability that *at least one* of is prime is bounded uniformly from below. One then applies the Bergelson intersectivity lemma to those events and uses many applications of the pigeonhole principle to conclude.

## 39 comments

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26 January, 2023 at 11:24 am

Anonymous“this answers a question of >>ErdÅ‘s”, Erdős is misspelled

[Corrected, thanks – T.]26 January, 2023 at 5:29 pm

AnonymousConsidering theorem 1(iii), what is the strongest known necessary or sufficient conditions for the sequence for the conclusions of theorem 1 ?

[Sorry, I am not able to interpret this question. -T]27 January, 2023 at 2:00 am

remyI propose to take a side step or look at the integers in different ways. For that I propose to introduce all the primes less than the value of the integer and to look only at the vector of all the modulos.

the vector is decomposable in sum ->glodbach’s conjecture

and that must also, can be , simplify your approach

Sorry, I am not able to interpret this question.

remy

27 January, 2023 at 2:49 am

RaphaelSorry, maybe too naive question, but can we give some explicit initial elements of any example for the two sequences?

1 February, 2023 at 10:04 am

Terence TaoUnfortunately not (though we can of course normalize one of the elements to be anything we like by translating the other sequence in the opposite direction). Even the simpler problem of producing an explicit positive integer such that are prime for infinitely many is open; we know that there is at least one such with , and every even integer should in fact have this property (de Polignac’s conjecture), but we cannot exhibit one explicitly. (Note that any difference of two elements of the first sequence will be of this form.)

1 February, 2023 at 11:59 pm

RaphaelVery interesting, thank you very much! Follow-up: Is there already a conjecture that for every $ latex n $ there is an infinite sequence of prime- tupels (with an appropriate unique definition of prime- tupel, generalizing twins, triples, …)?

27 January, 2023 at 7:15 am

Yemon ChoiDoes Theorem 1(ii) answer this question which I raised years ago on MathOverflow? https://mathoverflow.net/questions/3347/is-the-set-of-primes-translation-finite

(The original context was related to weak compactness / weak almost periodicity, see Defn 2.3 + Thm 2.4 + Lemma 2.5 of https://arxiv.org/abs/0811.4432 )

1 February, 2023 at 10:07 am

Terence TaoYes, it seems that Theorem 1.5 of our paper is precisely the assertion that the primes are not translation-finite. I’ll add these references to the next revision of the paper.

27 January, 2023 at 10:56 am

AnonymousIt seems that the formulation of theorem 1(III) is not sufficiently clear (because the primes are “positive density subset of the primes” – so the replacement of the primes by themselves should not change the conclusions !)

[“positive density” clarified to “positive relative density” -T.]27 January, 2023 at 3:55 pm

aHappy new year

28 January, 2023 at 5:12 am

Older but not wiserBeen 19 years since I took grad real analysis. Forgot the arguments you posted. Yet this post is short and very understandable.

28 January, 2023 at 2:38 pm

AnonymousIs it possible to generalize these results to sumsets in ?

28 January, 2023 at 6:39 pm

Sr Sidney Silvaas resoluções onde ele estiver presente…. com relação aos números primos, onde um Brasileiro afirma cientificamente que alguns números não são primos como todos(as) vem relatando de tempos passados, com uma simples PA(Progressão Aritmética), desvendei seus enigmas, onde afirmo cientificamente que não são primos e nem primos gêmeos: segue os números relatados a seguir: 2; 19; 41; 59; 61; 79; 101; 139; 179; 181; 199; 239; 241; 281; 359; 401; 419; 421; 439; 461; 479; 499; 521; 541; 599; 601; 619; 641; 659; 661; 701; 719; 739; 761; 821; 839; 859; 881; 919; 941; 1019; 1021; 1039; 1061; 1181; 1201; 1259; 1279; 1301; 1319; 1321; 1361; 1381; 1399; 1439; 1459; 1481; 1499; 1559; 1579; 1601; 1619; 1621; 1699; 1721; 1741; 1759; 1801; 1861; 1879; 1901; 1979; e mais alguns…… sendo assim “A Hipótese de Rielmann perde totalmente sua força” o autor Sr Sidney Silva.

1 February, 2023 at 10:50 am

Terence TaoI would imagine that the tools used here (in particular, the Maynard sieve) would extend to other number fields, for instance establishing an analogous result for the Gaussian primes, but I have not checked all the details.

28 January, 2023 at 3:32 pm

AFOf the conjectures of Dickson and Hardy–Littlewood, neither is stronger than the other, because while Hardy and Littlewood do give an asymptotic formula, Dickson considers general linear functions (where the leading coefficient, ie. the step size is allowed to vary).

2 February, 2023 at 8:00 pm

Keiju SonoIn page 3, the fifth line of the proof of Prop2.1

j>1 will be j>i.

I think the question of how small the width h_{k}-h_{1} of the prime producing k-tuple in Theorem 1.5 can be is one of the essential questions.

Have you considered this?

[Thanks, this typo will be corrected in a future revision of the ms. The most recent update of the paper now contains some discussion of growth rates; see Remark 3.3. -T.]5 February, 2023 at 2:16 am

ArmanPlease Prof creareate a YouTube channel and you will get more than one billion subscribers please

12 February, 2023 at 2:46 pm

AnonymousIt will result in anger from other math agencies that have benefited from having his talk on their channel on Youtube. It’s not advised.

5 February, 2023 at 2:33 am

ArmanProf share more if you make a video will be much clearer but if you have not time post more i am very huge fan of MATH)

11 February, 2023 at 10:22 am

ducduc2710Some calculations involving prime numbers https://www.researchgate.net/publication/367227821_Certaines-des-conseuquenceus-suivantes

11 February, 2023 at 4:42 pm

RhmnI saw your question on MO whose answer by Joel sharpens your result on uncountability of X! Can’t we simply argue by saying that X is an uncountable compact subspace of Cantor space so it is of size continuum?

[Yes, this also works, though Joel’s argument does not require the knowledge that X is compact. – T.]12 February, 2023 at 2:14 pm

aHappy Feb 12

13 February, 2023 at 7:16 pm

wangweiProof of Four color theorem

Suppose there is a map that can only be colored in five colors. Remove a country. If it can only be colored in five colors after removal, then remove the country. Otherwise, leave it. Each country will perform and operations in turn until all countries have completed one round, and then the next round will perform the same operations until no country has been removed in one round. In this way, the new map formed in the remaining countries can only be colored in five colors, and if any country is removed, it can be colored in four colors.

In this way, the neighboring countries of each country must contain four colors, because if a country contains only three colors, it can be colored with four colors after removing the country, then it can also be colored with four colors without removing the country (new map), which is inconsistent with the new map.

If the neighboring countries of each country contain four colors, the new map should only be colored with five colors after removing one country, which is contradictory, so the assumption is not tenable. There is no map that can only be colored with five colors.

Is my proof correct？

14 February, 2023 at 9:59 am

AnonymousIt is not clear why your proof can’t be applied to another numbers (e.g. 3 0r 5) than 4 (in other words. it is not clear from your proof what is special about 4)

21 February, 2023 at 10:11 am

Anonymous4 is very special in analytic number theory.

22 February, 2023 at 3:32 pm

Sr Sidney SilvaPrezado nobre amigo, interessante sua pergunta, pois com uma simples PA(Progressão Aritmética) padronizei duas fórmulas: vejamos como ficaria pegar um número primo e dividir ou seja usar a fatoração de números primos para ser fatorado do menor para o maior e do maior para o menor, assim será considerado um número primo:

esse número é primo 12280616105497 e somente pode ser dividido e fatorado somente pelos números 1867; 1871; 1873; 1877 que também são primos vejamos como fica minha tese:

do menor para o maior:

12280616105497 1867

6577726891 1871

3515621 1873

1877 1877

1

Agora irei fatorar com números primos do maior para o menor:

12280616105487 1877

6542683061 1873

3493157 1871

1867 1867

1

Provando minha tese que o numeral 2, 19 e outros já citados acima não são primos, e os primos gêmeos não existe. lendo este número primo teremos; (Doze Trilhões, Duzentos e Oitenta Bilhões, Seiscentos e Dezesseis Milhões, Cento e Cinco Mil Quatrocentos e Oitenta e Sete.), com esta simples fórmula cheguei na casa dos 92 Nonilhões, e é Finito…. Sr Sidney Silva.

23 February, 2023 at 3:32 am

Luis Martinez12280616105497 = 3515621*3493157

is not prime

23 February, 2023 at 2:52 pm

Sr Sidney SilvaSim, são primos, agora se tiver usando o numeral dois e muitos outros que mencionei que não são primos pode ser que sua afirmação esteja correta, porém todos que fatorei são primos tanto o de 14 dígitos e seus resultados são primos, prova cientifica minha tese esta correta. Sr Sidney Silva

24 February, 2023 at 1:04 am

remyI specify that there is not a prime number at each

different primorielle

an example with a series of twin primes

m=11590571 = 11590571

mm=m+2 = 11590573

m%(2*3*5*7*11*13*17) = 359351= 43 * 61 * 137

mm%(2*3*5*7*11*13*17) = 359353

m%(2*3*5*7*11*13) = 29021

mm%(2*3*5*7*11*13) = 29023

m%(2*3*5*7*11) = 1301

mm%(2*3*5*7*11) = 1303

m%(2*3*5*7) = 41

mm%(2*3*5*7) = 43

m%(2*3) = 5

mm%(2*3) = 1->3

19 February, 2023 at 3:52 am

Luis MartinezKnow primes number (included in sequences)

Hence the Mersenne primes are verified in the form:

if

.

With

20 February, 2023 at 1:13 pm

Sr Sidney SilvaNeste enunciado o 19 não é primo, então cai por terra esta hipótese!!

23 February, 2023 at 5:06 am

remyyour conjecture is true because,for me

(largePrimeNum)mod(2*3*5*7*11*…101)=PrimeNum

(largePrimeNum)mod(2*3*5*7*11*…97)=PrimeNum

…

(largePrimeNum)mod(2*3*5*7*11*…89)=PrimeNum

(largePrimeNum)mod(2*3*5*7*11*…83)=PrimeNum

…

(largePrimeNum)mod(2*3*5*7*11*…47)=PrimeNum

(largePrimeNum)mod(2*3*5*7*11*…43)=PrimeNum

…

(largePrimeNum)mod(2*3*5*7*11)=PrimeNum

(largePrimeNum)mod(2*3*5*7)=PrimeNum

for the demonstration it is necessary to write the prime numbers with a primorielle base

Attached is a file with the first 1000 prime numbers in primorielle base

https://www.cjoint.com/doc/23_02/MBvp6pGpfsJ_sortie.txt

so

largePrimeNum=pq+r=primorielle n+prime

(largePrimeNum)mod(primorielle )=prime

it is a consequence of the proposal of the available demonstrations

Click to access 2206.0068v1.pdf

and don’t repeat it but chatGPT doesn’t know how to make a small prime number from a large prime number

thank you for your attention

24 February, 2023 at 1:05 am

remyI specify that there is not a prime number at each

different primorielle

an example with a series of twin primes

m=11590571 = 11590571

mm=m+2 = 11590573

m%(2*3*5*7*11*13*17) = 359351= 43 * 61 * 137

mm%(2*3*5*7*11*13*17) = 359353

m%(2*3*5*7*11*13) = 29021

mm%(2*3*5*7*11*13) = 29023

m%(2*3*5*7*11) = 1301

mm%(2*3*5*7*11) = 1303

m%(2*3*5*7) = 41

mm%(2*3*5*7) = 43

m%(2*3) = 5

mm%(2*3) = 1->3

24 February, 2023 at 12:57 pm

Sr Sidney SilvaPrezado Srs(as), prove que esses números que mencionei pode ser encontrado números gêmeos, não vão encontrar, pois dentro da lista que me foi passado, aparece o 2, 19,41, 59, 61, 79 e muitos outros que citei não ser primos, enfim esses números não são primos, gostaria que todos(as) revesse seus conceitos, pois existe argumento dentro de um embasamento que esses números não são primos, os que citei acima em epigrafe.

24 February, 2023 at 1:02 pm

Sr Sidney SilvaNobre amigo, ficou vago sua explanação, prove sem esses números que apresentei que não são primos para ver se seu resultado esta correto? tenta olhar de outro ângulo minhas teorias… pois minhas teses estão de acordo com a revista científicas Sidney Silva.

27 February, 2023 at 12:18 am

remyI present a prime number generator based on an arithmetic progression (a sum) and I justify this progression with a demonstration proposal.

I do not change the rules of the game

27 February, 2023 at 2:47 pm

Sr Sidney SilvaPrezado nobre amigo interessante sua explanação com meu respeito a todos(as) aqui presente neste singelo canal, tenho um conceito sobre o que é números primos:

“Pra ser um número primo terá que dividir somente pelo número primo, sendo do menor para o maior, e do maior para o menor, só assim poderá ser um número primo exato e finito” essa Lei é aplicada no momento em que for feita a Fatoração de números primos”…. ao contrário que todos(as) vem relatando em tempos passados, usando uma teoria já obsoleta para os tempos atuais, Sr Sidney Silva. Porém não é possível esta sequência, pois tem alguns números que não são primos e os primos gêmeos não existe,

Estes números que irei apresentar não são primos!!

2; 19; 41; 59; 61; 79; 101; 139; 179; 181; 199; 239; 241; 281; 359; 401; 419; 421; 439; 461; 479; 499; 521; 541; 599; 601; 619; 641; 659; 661; 701; 719; 739; 761; 821; 839; 859; 881; 919; 941; 1019; 1021; 1039; 1061; 1181; 1201; 1259; 1279; 1301; 1319; 1321; 1361; 1381; 1399; 1439; 1459; 1481; 1499; 1559; 1579; 1601; 1619; 1621; 1699; 1721; 1741; 1759; 1801; 1861; 1879; 1901; 1979; Prove que esta Tese esta errada

cientificamente… Sr Sidney Silva.

Irei relatar uma prova de que minha Tese esta correta, vejamos como fica:

pois com uma simples PA(Progressão Aritmética) padronizei duas fórmulas: vejamos como ficaria pegar um número primo e dividir ou seja usar a fatoração de números primos para ser fatorado do menor para o maior e do maior para o menor, assim será considerado um número primo:

esse número é primo 12280616105497 e somente pode ser dividido e fatorado somente pelos números 1867; 1871; 1873; 1877 que também são primos vejamos como fica minha tese:

do menor para o maior:

12280616105497 1867

6577726891 1871

3515621 1873

1877 1877

1

Agora irei fatorar com números primos do maior para o menor:

12280616105487 1877

6542683061 1873

3493157 1871

1867 1867

1

Provando minha tese que o numeral 2, 19 e outros já citados acima não são primos, e os primos gêmeos não existe. lendo este número primo teremos; (Doze Trilhões, Duzentos e Oitenta Bilhões, Seiscentos e Dezesseis Milhões, Cento e Cinco Mil Quatrocentos e Oitenta e Sete.), com esta simples fórmula cheguei na casa dos 92 Nonilhões, e é Finito…. Sr Sidney Silva. dentro deste valor encontrado multipliquei 1867*1871*1873*1877= 12280616105497 todos são números primos, irei citar outro exemplo mais simples: Sidney Silva

o primeiro numero primo seria o 3 vejamos como ficaria minha tese e teoria:

3*5*7*11*13*17= 255255 vejamos como fica pela minha definição de números primos:

Do menor para o maior:

255255 3

85085 5

17017 7

2431 11

221 13

17 17

1

Agora do maior para o menor:

255255 17

15015 13

1155 11

105 7

15 5

3 3

1

Prova que os valores fatorados somente com números primos; será considerado números primos, pois foi dividido somente com os números primos existente dentro da minha tese e teoria…porém haverá um único numero primo que será dividido por ele mesmo e o resultado será 1, classificando como os verdadeiros números primos………esse é meu conceito e a definição para os números primos…….meu cordial abraço Sr Sidney Silva.

28 February, 2023 at 12:58 am

remynão concordaremos, lamento,a direcção do factoring não altera o resultado do factoring

24 February, 2023 at 2:47 am

remyAttached is a file with twin primes that shares some of their writing

https://www.cjoint.com/doc/23_02/MBykOnigb1J_sortie.txt

I recommend using notepad++ there is no automatic line wrap