Some works on the Generalized Andrica's Conjectures and Variants
Etudes sur la conjecture d'Andrica généralisée et certaines variantes
Abstract
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
Domains
Mathematics [math]
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