- #1

pedja

- 15

- 0

Consider the following array of natural numbers :

[tex]\begin{array}{ccccccccc}

1 & 2 & 4 & 7 & 11 & 16 & 22 & 29 & \ldots \\

3 & 5 & 8 & 12 & 17 & 23 & 30 & 38 & \ldots \\

6 & 9 & 13 & 18 & 24 & 31 & 39 & 48 & \ldots \\

10 & 14 & 19 & 25 & 32 & 40 & 49 & 59 & \ldots \\

15 & 20 & 26 & 33 & 41 & 50 & 60 & 71 & \ldots \\

21 & 27 & 34 & 42 & 51 & 61 & 72 & 84 & \ldots \\

\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &

\end{array}[/tex]

**Question :**Are there infinitely many primes in every row of this array ?

My attempt :

The nth term of the first row is given by :

[tex]a_n=\frac{1}{2}(n^2-n+2)[/tex]

[tex]\text{for}~ n=2k~\text {we have :} [/tex]

[tex]a_{2k}=P(k)=2k^2-k+1[/tex]

[tex]\text{and for}~ n=2k-1~\text{ we have :}[/tex]

[tex]a_{2k-1}=Q(k)=2k^2-3k+2[/tex]

Note that both P(k) and Q(k) are irreducible over integers .

Also note that : [tex]\gcd(P(1),P(2),\ldots)=1 ~\text{and}~ \gcd(Q(1),Q(2),\ldots)=1[/tex]

So, according to Bunyakowsky conjecture both P(k) and Q(k) generates for natural arguments infinitely many prime numbers . Therefore , if Bunyakowsky conjecture is true first row contains infinitely many primes . One can draw same conclusion for all other rows .

Is my reasoning correct ? Is there some other approach to this problem ?