From HYPOTHESIS H AND AN IMPOSSIBILITY THEOREM OF RAM MURTY.
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