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Pré-Publication, Document De Travail Année : 2020

Some works on the Generalized Andrica's Conjectures and Variants

Etudes sur la conjecture d'Andrica généralisée et certaines variantes

Lionel Laurore

Résumé

The Andrica’s conjecture affirm that for every prime number √p(n+1) -√p(n) <1 and more precisely, that the upper bound of this function is reached for pn = 7. Another conjecture, called generalized Andrica’s conjecture, affirm that, for every prime number, there is an only real number x, such as : p(n+1)^x – p(n)^x = 1, and the minimum value of x=x(p(n)( is obtained for p(n) = 113. Florentin Smarandache in 2000 , has listed a set of conjectures linked to Andrica’s conjecture such as p(n+1)^(1/k) – p(n)^(1/) < 1/k for all pn and all k>2. David Lowry-Duda in 2017 [4], has formulated the following open question: For any constant C, what is the smallest x such as p(n+1)^x – p(n)^x = C . In 2018 Matt Wisser from Victoria University of Wellington has proposed a variant of Andrica’s conjecture and presents an original approach to prove that Ln^k(p(n+1)) - Ln^k(p(n)) has an upper bound and gives some numerical values for k≤5. In the present work, we are going to establish a set of results regarding: Part 1 : the generalized Andrica’s conjecture : Fn(x) = p(n+1)^x – p(n)^x = 1. We shall build some lower and upper bound functions X-(p(n)) et X+(p(n)) such as, for all p(n) , X-(p(n)) ≤ x(p(n)) ≤ X+(p(n)). We shall study the behavior of these functions X-(p(n)) et X+(p(n)) and show under a weaker condition than Cramer’s conjecture that x is minimal for p(n) = 113. We’ll give the proof that p(n+1)^(1/k) – p(n)^(1/k) < 1/k for all k > 2. We’ll show that all x < 1, Fx(p(n)) = p(n+1)^x – p(n)^x has an upper bound for all xͼ[0,0.475[ and Cramer’s conjecture for all xͼ[0.475,1[. Part 2 : We propose an alternative approach to prove that for all integer k, Lk(p(n)) = Ln^k(p(n+1)) - Ln^k(p(n)) has an upper bound and we’ll build some lower and upper bound functions Lk-(pn) and Lk+(p(n)). We’ll try to estimate the integer sequence p(n,k) such as Maxn (Lk(p(n))) = Lk(p(n,k)). We’ll propose some upper bound values of Lk(p(n)) for all k<25. We’ll deduce that for all integers k>0 and r>0, Gk,r(p(n)) = Ln^k(p(n+1)^r) - Ln^k(p(n)^r) has an upper bound value. We’ll prove that Ly(p(n)) = Ln^y(p(n+1)) – Ln^y(p(n))=1 has a solution y(p(n))ͼℝ for all p(n) and we will propose an approximation of y(p(n)). Part 3 ; We’ll generalize the different results obtained in part 1 and 2 to establish that the following functions on prime number accept upper values : Hk,x(p(n)) = p(n+1)^x.Ln^k(p(n+1)) - p(n)^x.Ln^k(p(n)) for all integer k>0 and real 0≤x<1 We’ll study the set of (x,y)∈ℝxℝ when the functions Hx,y(p(n)) = p(n+1)^x.Ln^y(p(n+1)) - p(n)^x.Ln^y(p(n)) is bounded. We’ll try to state the set of solution (x,y) which verify Hx,y(p(n)) = 1
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hal-02560395 , version 1 (27-05-2020)

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Lionel Laurore. Some works on the Generalized Andrica's Conjectures and Variants. 2020. ⟨hal-02560395⟩
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