# Find quadratic polynomial such that f(x)=x^2+ax+b is prime

a. Find quadratic polynomial f(x)=x²+ax+b , a,b ∈ℤ such that f(x) is prime for 1≤n≤40

I know that the polynomial f(x)=x²-x+41 takes prime values for all 1≤n≤40. But how can I find this polynomial with sage?

b. Find k (as large as possible) and quadratic polynomial such that f(x)=x²+ax+b (a,b ∈ℤ) is prime for 1≤n≤k

I only find can b if I know a. But how can I find both?

P=Primes();
for b in range (0,1000000):
success=true;
for x in range (1,41):
if not (x^2-x+b) in P:
success=false;
break;
if success:
print b;

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This looks like homework. If you want some help, you should ask more precise questions related to your research in solving those exercises, especially where you are locked.

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Homeworks tend to lose interest after some weeks, because solutions are provided... So i was waiting some time...

Here is a piece of code, which - given a lower bound $k$, say $k=70$, - is searching for polynomials of the shape $$f(X) =X^2 +aX+p\ ,$$ so that $f(j)$ is a prime number for all $j$ in the interval from $0$ top $k$. In particular, the numbers $f(0)=p$ and $f(1)=1+a+p=q$ are prime numbers. And these numbers are the strarting point for the search.

The tuple $(p,q)$ determines, $a$, so also the polynomial $f$, and we check in all cases if the needed values $f(0)=p$, $f(1)=q$, $f(2)$, ... , $f(k)$ are prime numbers. If yes, we record this polynominal. The search is done by searching in a fixed range PRIMES for the first two primes $p,q$.

Code:

FIRST_TWO_PRIMES_UPPER_BOUND = 10000
var('X');

k      = 70
count  = 0
pols   = []
PRIMES = list(primes(FIRST_TWO_PRIMES_UPPER_BOUND))

for p in PRIMES:
count += 1
if count % 100 == 0:
print(f"count = {count} :: checking prime {p}...")
# we are searching for a polynomial of the shape x^2 + ax + p, a>0, with prime values on [0..k]
for q in PRIMES:
q_works = True # so far
a = q - p - 1
for j in [2..k]:
if not ZZ(j^2 + a*j + p).is_prime():
q_works = False
# print(p, 'FAILS FOR', q, a, j, (j^2 + a*j + p).factor())
break
if q_works:
pol =  X^2 + a*X + p
print(f"!!! Found polynomial: {pol} !!!")
pols.append(pol)

if pols:
print(f"SOLUTIONS FOR k = {k}:")
for pol in pols:
print(pol)
else:
print(f"NO SOLUTIONS FOR k = {k}:")


And here are the found solutions for $k=70$:

SOLUTIONS FOR k = 70:
X^2 - 61*X + 971
X^2 - 63*X + 1033
X^2 - 65*X + 1097
X^2 - 67*X + 1163
X^2 - 69*X + 1231
X^2 - 71*X + 1301
X^2 - 73*X + 1373
X^2 - 75*X + 1447
X^2 - 77*X + 1523
X^2 - 79*X + 1601


The last polynomial, $X^2 -79X +1601=X(X-79)+1601$ takes the same values in the points $j$ and $79-j$, so we obtain prime values (with repetitions) even for all $j$ from $0$ to $79$.

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