Acta Universitatis Danubius. Œconomica, Vol 13, No 3 (2017)
Œconomica
Again about Andrica’s Conjecture...
Cătălin Angelo Ioan^{1}
Abstract: The paper establishes an equivalence of the Andrica’s conjecture in the direction of an increase of the difference of square root of primes by a combination of two consecutive primes.
Keywords: Andrica’s conjecture; prime
JEL Classification: C002
1. Introduction
In a previous paper, entitled “About Andrica’s conjecture” the authors have established an equivalence of conjecture Andrica by considering the ratio of two consecutive prime numbers. Because the average deviation calculated relative to the two terms, in this article will study another limit for the difference of square roots of two consecutive prime numbers.
A number pN, p2 is called prime number if its only positive divisors are 1 and p.
Even if do not know much about prime numbers, there exist a lot of attempts to determine some of their properties, many results being at the stage of conjectures.
A famous conjecture relative to prime numbers is that of Dorin Andrica. Denoting by p_{n}  the nth prime number (p_{1}=2, p_{2}=3, p_{3}=5 etc.), Andrica’s conjecture ([1]) states that:
n1
In [3] we have found the following:
Theorem
Let p_{n} the nth prime number. The following statements are equivalent for n5:

;

0 such that: .
In the following, we shall prove a stronger theorem of equivalence of Andrica’s conjecture.
2. Main Theorem
Theorem
Let p_{n} the nth prime number. The following statements are equivalent for n5:

;

0 such that: .
Proof
First of all let the function f:[e,)R, . We have: 0 therefore f is a strictly decreasing function. For n2 we have therefore: that is: 1.
21 Because <1 follows that: 0 < =1.
12 If we take the logarithm in the relationship, it becomes:
.
Let now the function:
g:(a,)R, with a>2
We have now: =
Because the denominator of g’ is positive, we must inquire into the character of the function:
h:(a,)R, .
Computing the derivative of h:
Let now:
y:(a,)R,
and the derivative:
Let now the function (the numerator of y):
and, also, the derivative:
Because xa we have that 0 therefore z is a strictly increasing function.
But 0 therefore 0 xa.
In this case 0 then y is also a strictly increasing function.
But = .
The function has 0 for x2 therefore u is decreasing and = 0.
We have now: 0 therefore 0 for xa+1.
Now 0 which give us: h is increasing.
But 0 therefore 0 for xa+1.
Because now: 0 implies that g is increasing and with we find that g(x)0 xa+1.
From hypothesis 1 (Andrica’s conjecture), we have: and noting: x= , a= we have: x(a,a+1) therefore:
Considering = 0, the statement 2 is now obvious. Q.E.D.
3. Determination of the Constant
Using the Wolfram Mathematica software, in order to determine the constant (for the first 100000 prime numbers):
Clear[“Global`*”];
numberiterations=100000;
minimum=1000;
k=0;
For[i=5,i<=numberiterations+4, i++,
difference=Sqrt[Prime[i+1]]Sqrt[Prime[i]];
ratio=Log[Prime[i+1]]*Prime[i]Log[Prime[i]]*Prime[i+1];
log=Log[difference]/ratio;
If[log<minimum,minimum=log]];
Print["Minimum=",N[minimum,1000]]
we found that the first 1000 decimals are:
=0.001801787909180184090558881990879581852587815188626060829671181995518153228056185868661669722893637929905150138361741357987598217520912492958000134271102248291291440100211921382959611030962352046213123107738700539021075748371514085755924571808071605072827284127643779109598663531522374100243861797823777482028364380170936681469375189124611595038701054740899835310850858481264555165634252199160623380073272834451080219681979931918287609129486097360176969992548676629716572067527720901123119401797627368003734134881964963643241047796485654858914187103720570510400193723300037859727351479951565306627468352075884099806617621474175589423220844469527382500914548671086635285509959540990596065572675463044441151661992941475164500380927975560830750502367458528934167921925547374264915121574707114622773865335533852158934313781909407119398388818028233946073756228798804604974623153893100857242848052370682767307818661501668704604756723146711562022353266081970578858543065045549989697839196705824350226507331762.
4. References
Andrica, D. (1986). Note on a conjecture in prime number theory. Studia Univ. BabesBolyai Math. 31(4), pp. 44–48.
Guy, R.K. (1994). Unsolved Problems in Number Theory. Second Edition. Springer Verlag, New York.
Ioan, C.A. & Ioan, A.C. (2015). About Andrica’s conjecture. Acta Universitatis Danubius Œconomica, no 1, vol. 11, pp. 149153.
Wolf, M. (2010). A Note on the Andrica Conjecture. arXiv:1010.3945v1 [math.NT] 19 Oct.
1 Associate Professor, PhD, Department of Economics, Danubius University of Galati, Romania, Address: 3 Galati Blvd., Galati 800654, Romania, Tel.: +40372361102, Corresponding author: catalin_angelo_ioan@univdanubius.ro.
AUDŒ, Vol. 13, no. 3, pp. 207212
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