From [HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY](http://alpha.math.uga.edu/~pollack/impossibility.pdf).


On p. 13



> 
> BUNYAKOVSKY’S CONJECTURE ( WEAK FORM ). Let $f$ be a polynomial with
>  integer coefficients and positive leading coefficients which 
>  is irreducible over $\mathbf{Q}$. Let $d := \gcd\{ f(n)\}\_{n \in \mathbf{Z}} $.
>  Then $f (n)/d$ is prime for at least one positive integer $n$.
> 
> 
> 


On p. 6:



> 
> BUNYAKOVSKY’S CONJECTURE. Let $f$ be a polynomial with
>  integer coefficients and positive leading coefficients which 
>  is irreducible over $\mathbf{Q}$. Let $d := \gcd\{ f(n)\}\_{n \in \mathbf{Z}} $.
>  Then $f (n)/d$ is prime for infinitely many positive integers
>  $n$.
> 
> 
> 


The standard formulation of Bunyakovsky's conjecture requires $d=1$.


We claim explicit counterexample to both.


Let $f$ be the degree $12$ polynomial:



```
f=x^12 - 197*x^11 + 16976*x^10 - 859146*x^9 + 28693351*x^8 - 669930367*x^7 + 11254469458*x^6 - 137432686432*x^5 + 1213054497367*x^4 - 7558495778147*x^3 + 31594161981276*x^2 - 79613982402450*x + 91528191555876

```

$$f=x^{12} - 197 x^{11} + 16976 x^{10} - 859146 x^{9} + 28693351 x^{8} - 669930367 x^{7} + 11254469458 x^{6} - 137432686432 x^{5} + 1213054497367 x^{4} - 7558495778147 x^{3} + 31594161981276 x^{2} - 79613982402450 x + 91528191555876$$


$f$ is irreducible over $\mathbf{Q}[x]$. $\gcd(f(1),f(7))=6$, so
$d \le 6$ and by congruence arguments $d=6$.


Let $N=546=2 \cdot 3 \cdot 7 \cdot 13$.


For all natural, $x$ we claim $\gcd(f(x)/6,N) > 1$.
$f(x)/6$ is periodic modulo $N$, so must check only $x \in [1,N]$.


By fast computation, $\gcd(f(x)/6,N) > 1$ for $x \in [1,N]$.


So $f(x)/6$ can prime only if it equals the primes factors of
$N$ at positive integers, which is not possible even at rationals.



> 
> Q1 What is wrong with this alleged counterexample?
> 
> 
> 


In case of positive answer:



> 
> Q2 From it, can we can counterexample with $d=1$, which is the
>  usual formulation of Bunyakovsky's conjecture.
> 
> 
> 


Searching the web, couldn't find how to contact the author of the paper.



> 
> Q3 How to contact the author of the paper via email?
> 
> 
> 


Got numeric support in both sage and pari.


Trying to follow the proof of the reformulation, we couldn't
find $A,B$ with the desired properties.


Sage verification code:



```
def bunyakovskweakyani1():
    """
    """
    K.<x>=QQ[]
    f=x^12 - 197*x^11 + 16976*x^10 - 859146*x^9 + 28693351*x^8 - 669930367*x^7 + 11254469458*x^6 - 137432686432*x^5 + 1213054497367*x^4 - 7558495778147*x^3 + 31594161981276*x^2 - 79613982402450*x + 91528191555876
    N=546
    print 'factor(f)',factor(f)
    d=gcd(ZZ(f(1)),ZZ(f(7)))
    gg=gcd([ZZ(f(i)) for i in xrange(N)])
    print 'gcd(f(1),f(7))=d=',d,'g=',gg
    print '  should not raise error'
    for i in xrange(N):
        a=ZZ(f(i)/d)
        g=gcd(a,N)
        if g == 1:
            print 'g == 1',i
            assert False,'coprime to N'
    print '  no error, all roots should be non-integers'
    for p,_ in factor(N):
        ro=(f-p).roots(multiplicities=False)
        print p,'roots=',ro

```


What is wrong with this counterexample to the Weak Bunyakovsky's conjecture and reformulation of Bunyakovsky's conjecture?
$f(x)/6 + N\mathbb Z$ is *not* periodic modulo $N$. It is periodic modulo $6N$, so you have to check a larger range. And indeed, $\gcd(f(637)/6,N)=1$. 


