Acta Universitatis Danubius. Œconomica, Vol 11, No 2 (2015)
A Conjecture Concerning Prime Numbers
Cătălin Angelo Ioan^{1}, Alin Cristian Ioan^{2}
Abstract. The paper examines the conditions under which a second degree polynomial generates primes variable values between 0 and the constant term  1. It is shown that for values of the constant term equal to or less than 41, such polynomials exist and we are proposing a conjecture that for the polynomials with the constant term greather than 41 the statement is not true.
Keywords: prime; polynomials
1. Introduction
The prime number theory dates back to ancient times (see the Rhind papyrus or Euclid's Elements).
A number pN, p2 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 (all unsuccessful) to determine it.
Unfortunately, many results about primes are at the stage of conjectures (theorems that seem to be valid, but remained unproven yet).
2. Main Results
Let , a,b,cZ, c2, .
We propose the determination of a,b,cZ such that =prime n= .
Using the Wolfram Mathematica software, in order to determine the polynomials (for c41):
Clear[“Global`*”];
polynomial[a_,b_,c_,n_]=a*n^2+b*n+c;
lowerlimitc=2;
upperlimitc=41;
upperlimit=200;
For [c=lowerlimitc,c<=upperlimitc,c++,
maximuma=0;
maximumb=0;
maximumc=0;
maximumcounter=1;
increase=2*upperlimit;
If[PrimeQ[c],
For[a=upperlimit,a<=upperlimit,a++,
For[b=upperlimit,b<=upperlimit,b++,
counter=0;exit=0;
If[a0,For[n=0,n<c,n++,
If[PrimeQ[polynomial[a,b,c,n]]&&exit=0,counter=counter+1,exit=1]]];
If[exit=0&&counter maximumcounter,If[(Abs[a]+Abs[b])< increase,maximumcounter=counter;maximuma=a;maximumb=b;
maximumc=c;increase=Abs[a]+Abs[b]]]]]];
If[maximumcounter2, Print[“a=“,maximuma,” b=“,maximumb,” c=“,maximumc];
For[n=0,n<c,n++,Print[polynomial[maximuma,maximumb,maximumc,x]]],If[PrimeQ[c],Print[“c=“,c,” accepts no polynomial”]]]]
we get:
c 

Module of Primes 
2 

2,3 
3 

3, 3, 5 
5 

5, 5, 7, 11, 17 
7 

7, 7, 11, 19, 31, 47, 67 
11 

11, 11, 13, 17, 23, 31, 41, 53, 67, 83, 101 
13 

13, 19, 37, 67, 109, 163, 229, 307, 397, 499, 613, 739, 877 
17 

17, 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257 
19 

19, 19, 23, 31, 43, 59, 79, 103, 131, 163, 199, 239, 283, 331, 383, 439, 499, 563, 631 
23 

23, 23, 29, 41, 59, 83, 113, 149, 191, 239, 293, 353, 419, 491, 569, 653, 743, 839, 941, 1049, 1163, 1283, 1409 
29 

29, 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597 
31 

31, 73, 109, 139, 163, 181, 193, 199, 199, 193, 181, 163, 139, 109, 73, 31, 17, 71, 131, 197, 269, 347, 431, 521, 617, 719, 827, 941, 1061, 1187, 1319 
37 

37, 109, 173, 229, 277, 317, 349, 373, 389, 397, 397, 389, 373, 349, 317, 277, 229, 173, 109, 37, 43, 131, 227, 331, 443, 563, 691, 827, 971, 1123, 1283, 1451, 1627, 1811, 2003, 2203, 2411 
41 

41, 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601 
For c43 we have that from the condition that =prime n= implies that a+b=even. Indeed, if the prime is different from 2, we have that: =even for any n= except possibly two values for which =2.
But for two consecutive numbers n, n+1 such that =prime_{1}2, =prime_{2}2 we have that: , therefore the difference: that is .
The structure of the polynomial becomes then:
, a,b,dZ
Investigating for values of c greater than 41 to 1000, we have not found polynomials with this property, considering reasonable limits for a and b less than 200.
Finally we state the following:
Conjecture
There is not a polynomial , a,b,cZ, c43, such that =prime n= .
3. References
Guy, R.K. (1994). Unsolved Problems in Number Theory. Second Edition. New York: Springer Verlag.
Ioan, C.A. (2004). Theory of prime numbers  art and science. EuroEconomica, Nr. 9, pp. 5963.
1 Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, Fax: +40372 361 290, Corresponding author: catalin_angelo_ioan@univdanubius.ro.
2 Nicolae Oncescu College, Braila, Address: 13 ŞOS. Brăilei, City: Ianca, Brăila County, Tel.: +40239668 494, Email: alincristianioan@yahoo.com.
AUDŒ, Vol. 11, no. 2, pp. 242245
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