Acta Universitatis Danubius. Œconomica, Vol 11, No 1 (2015)

About Andrica’s Conjecture

Cătălin Angelo Ioan¹, Alin Cristian Ioan²

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 power of a ratio of two consecutive primes.

Keywords: Andrica’s conjecture; prime

JEL Classification: E23

1. Introduction

Even prime number theory dates back to ancient times (see the Rhind papyrus or Euclid's Elements) it retains its topicality and fascination to any mathematician due to numerous issues remain unresolved to this day.

A number pN, p2 is called prime if its only positive divisors are 1 and p. The remarkable property of primes is that any nonzero natural number other than 1 can be written as a unique product (up to a permutation of factors) of prime numbers to various powers.

If there is not a formula, for the moment, generating prime numbers, there exist a lot of attempts (many successful, in fact) to determine some of their properties.

Unfortunately, many results are at the stage of conjectures (theorems that seem to be valid, but remained unproven yet).

A famous conjecture relative to prime numbers is that of Dorin Andrica. Denoting by p_n - the n-th prime number (p₁=2, p₂=3, p₃=5 etc.), Andrica’s conjecture ([1]) states that:

n1

Noting g_n=p_n+1-p_n – the prime gap that is the difference between two successive prime numbers, the conjecture can be written in the form:

n1

The equivalence is immediate because  which is obviously equivalent to: .

Even if Andrica’s conjecture is weaker than that resulting from Oppermann's conjecture which states that the attempts to prove have not been successful to this moment.

In the following, we shall prove a theorem of equivalence of Andrica’s conjecture with another conjecture of increasing the difference of square root of consecutive primes with a power of their ratio.

2. Main Theorem

Theorem

Let p_n the n-th prime number. The following statements are equivalent for n5:

1. ;

2. 0 such that: .

Proof

21 Because <1 follows that: < =1.

12 If we take the logarithm in the relationship, it becomes:

  

.

Let now the function:

f:(a,)R, f(x)= with a>2

We have now: f’(x)= .

Noting g(x)= we get: g’(x)=ln x-ln a-ln(x-a)=h(x).

Because h’(x)= <0 we have that h is strictly decreasing. How h(a+1)=ln(a+1)-ln a>0, h(a+2)= <0, follows that (a+1,a+2) such that: h()=0 that is:

1. ln -ln a=ln(-a)

and after the monotony: h(x)>0 x< and h(x)<0 x>.

Because g’=h we obtain that g is strictly increasing on (a,) and strictly decreasing on (,).

But =0, =-, g()= and from (1):

g()=aln(-a)>0

It results that >>a+1 such that: g()=0 that is:

From monotonicity, we obtain that: g(x)>0 x(a,) and g(x)<0 x(,).

Therefore, f is strictly increasing on (a,) and strictly decreasing on (,).

As f(a+1)=0, f()= , and from (2): f()= >0, =1 it follows that: f(x)<0 x(a,a+1).

From hypothesis 1 (Andrica’s conjecture), we have: and noting: x= , a= we have: x(a,a+1).

Be so: = 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):

alfa:=1000.

For[n=5,n<100000,n++,alfa1=Log[Prime[n]/Prime[n+1],Sqrt[Prime[n+1]]-Sqrt[Prime[n]]];alfa=Min[alfa1,alfa]];

Print[N[alfa,1000]]

we found that the first 1000 decimals are:

=2.281103221027218229423232822443286599017584100009984588220198770755501695607955612552390209582835552197218929467815343425765174365517263629238806781607467589356396574145082119198576435873567859050606949765734239636479131860224037385179731071495942268916184068058269617751272081881749024559629325428694750309258404442895175344929226973898618839023663891240934504129986369574540734455985395956942690589958313834222509660041292032936969131875482708936809950924268932852699740905121223702115128603865911454535850929742715336117868971919273515459512287118318877766230702063182116913454786964284608231057854097720248263859745150386851610652371395957541613534650131084885714338495230915143345236023787960609518428948748046213447985903921468982086264170981653575011697562666536668586244503743588729350392068808501947089500551068277858220736778549760090107511452195156335525727194242907057388224636151079155629808656904777296450298019665316030034642716170057957313704524610621876879383627332120315454057552219

4. References

Andrica, D. (1986). Note on a conjecture in prime number theory. Studia Univ. Babes-Bolyai Math. 31 (4), pp. 44–48.

Guy, R.K. (1994). Unsolved Problems in Number Theory. Second Edition. New York: Springer Verlag.

Wolf, M. (2010). A Note on the Andrica Conjecture. arXiv:1010.3945v1 [math.NT] 19 Oct.

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