Problem:


Score : 800 points

Problem Statement

You are given a polynomial of degree N with integer coefficients: f(x)=a_Nx^N+a_{N-1}x^{N-1}+...+a_0. Find all prime numbers p that divide f(x) for every integer x.

Constraints

0 \leq N \leq 10^4

|a_i| \leq 10^9(0\leq i\leq N)

a_N \neq 0

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_N
:
a_0

Output

Print all prime numbers p that divide f(x) for every integer x, in ascending order.

Sample Input 1

2
7
-7
14

Sample Output 1

2
7

2 and 7 divide, for example, f(1)=14 and f(2)=28.

Sample Input 2

3
1
4
1
5

Sample Output 2

There may be no integers that satisfy the condition.

Sample Input 3

0
998244353

Sample Output 3

998244353



Sample input:
2
7
-7
14


Sample output:
2
7

Submission status: Wrong Answer

Solution code:
from functools import reduce
import random
import time
MOD = 59490679579998635868564022242503445659680322440679327938309703916140405638383434744833220555452130558140772894046169700747186409941317550665821619479140098743288335106272856533978719769677794737287419461899922456614256826709727133515885581878614846298251428096512495590545989536239739285290509870860732435109794045491640561011775744617218426011793283330497878491137999673760740717904979730106994180691956740996319139767941335979146924614249205862051182094081815447356600355657858936442589201739350474320966093143098768914226528498860747050474426418211313776073365498029087492308102876323562641651940555529817223923967479711293692697592800750806084802623966596666156361791014935954720420383555300213829134029560826306201326905674141090667002959274513478562033467485802590354539887133866434256562999666615074220857502563011098759348850149774083443643246907501430076141377431759984926248272553982527014011245310690986644461251576837779050124866641531488351016070300247918750613207952783437248236577750579092103591540592540838395240861443615842939673889972504776455114767522430347610500271801944701491874649306605557851914677316523034726044356368337787113190245601235705959816327016988315321850324065292964680148884818699916224411245926491625245571963741062587277736372090007125908660590314049976206281064703285728879581199596401313352724403492043120507597057004633720111008838673185885941472392623805512709896864520875740497617898566981218781313900004406341154884180472153171064617953661517880143988867189622824081729539392205701820144922307223627345876707465251206005262622236311161449785941834002795854996108322186230425179217618301526151712928790928933798369678844576216735378555233905935973195721247604933753363412045618703247367192610615234835041956551569232557323060407648325565048712478527583315981204846341857095134567470182330491262285172727037299211391244340592936174221176781260586188162350081192408213470101717320475998863222409777310443027421196981193126541663212124245716187453863438039402316877152286468198891603632606578778749292403571792687832081974134637014026451921536576338243322267400651083805535119025415817887075652758045539565968044552126338330231434466204888993650859153585380124240540573308417330478048240203241631072371322849430883727355239704116556046700749530006852187064160849175332758172150251213637470549781080491037088372092203085237973008861896576796238915011886636658033019385943299986285181723378096425117379056207797455963451889971988904466449319007760192467211209692128894691704353648198130409333996534250878389064152054828983825841234644875996912485916827004219887033833599723481903489316488764021700996686817244736947119285629049355809027206179193628292018441744552168286541735687924729198455883895791600170372255284216915442808139396541702893732917062958054499525549626455191658842064247047426187897146172971001949767308335268505414284088528125611263685734457560292833389995980698745893243832547007166243476192958601735336260255598581701267151224204461879782815468518040925292817115377696676461775120750971210951527384637825092221708015393564320979357698186262039029460777050248162599194429941464248920952161182024344007059684970270762248243899640259750891406957836740989312390963091260380990901672119736141666856448171380781556117025832312710039041398538035351795267732240730608951176127282191528457681241590895740457571038936173983449289126574141189374690274057472401359482497502067814596008557725079835212621242944853319496441084441343866446380876967613370281793088540430288658573281302876742323336651987699532240686371751448290351615451932054274752105688318766958111191822471878078268490672607804265064578569581247796205593441336042254502454646980009177290888726099355974892680373341371214123450598691915802122913613669446370194221846225597401405625536693874723700374068542217340621938985167525416725638580266200550048001242788847319217369432802469735271132107428204697172144851088692772696511622350096452418244968500432645305761138012204888798724453956720733374364721511323104353496583927094760495687785031050687852300161433757934364774351673531859394855389527851678630166084819717354779333605871648837631164630550613112327670353108353791304451834904017538583880634608113426156225757314283269948216017294095002859515842577481167417167637534527111130468710
sTime = time.time()

def gcd(a, b):
    b = abs(b)
    while b != 0:
        r = a%b
        a,b = b,r
    return a
def gcd_mult(numbers):
    return reduce(gcd, numbers)

N = int(input())
A = [int(input()) for _ in range(N+1)][::-1]
g = gcd_mult(A)

A = [a//g for a in A]

def f(n):
    ret = A[N]
    for i in range(N-1, -1, -1):
        ret = (ret * n + A[i]) % MOD
    return ret

ans = MOD
while time.time()-sTime < 1.5:
    k = random.randrange(10**10)
    ans = gcd(ans, f(k))

def primeFactor(N):
    i = 2
    ret = {}
    n = N
    if n < 0:
        ret[-1] = 1
        n = -n
    if n == 0:
        ret[0] = 1
    d = 2
    sq = int(n ** (1/2))
    while i <= sq:
        k = 0
        while n % i == 0:
            n //= i
            k += 1
            ret[i] = k
        if k > 0:
            sq = int(n**(1/2))
        if i == 2:
            i = 3
        elif i == 3:
            i = 5
        elif d == 2:
            i += 2
            d = 4
        else:
            i += 4
            d = 2
        
    if n > 1:
        ret[n] = 1
    return ret

for i in primeFactor(ans*g):
    print(i)

Submission status: Wrong Answer

Solution code:
from functools import reduce
import random
import time
MOD = 59490679579998635868564022242503445659680322440679327938309703916140405638383434744833220555452130558140772894046169700747186409941317550665821619479140098743288335106272856533978719769677794737287419461899922456614256826709727133515885581878614846298251428096512495590545989536239739285290509870860732435109794045491640561011775744617218426011793283330497878491137999673760740717904979730106994180691956740996319139767941335979146924614249205862051182094081815447356600355657858936442589201739350474320966093143098768914226528498860747050474426418211313776073365498029087492308102876323562641651940555529817223923967479711293692697592800750806084802623966596666156361791014935954720420383555300213829134029560826306201326905674141090667002959274513478562033467485802590354539887133866434256562999666615074220857502563011098759348850149774083443643246907501430076141377431759984926248272553982527014011245310690986644461251576837779050124866641531488351016070300247918750613207952783437248236577750579092103591540592540838395240861443615842939673889972504776455114767522430347610500271801944701491874649306605557851914677316523034726044356368337787113190245601235705959816327016988315321850324065292964680148884818699916224411245926491625245571963741062587277736372090007125908660590314049976206281064703285728879581199596401313352724403492043120507597057004633720111008838673185885941472392623805512709896864520875740497617898566981218781313900004406341154884180472153171064617953661517880143988867189622824081729539392205701820144922307223627345876707465251206005262622236311161449785941834002795854996108322186230425179217618301526151712928790928933798369678844576216735378555233905935973195721247604933753363412045618703247367192610615234835041956551569232557323060407648325565048712478527583315981204846341857095134567470182330491262285172727037299211391244340592936174221176781260586188162350081192408213470101717320475998863222409777310443027421196981193126541663212124245716187453863438039402316877152286468198891603632606578778749292403571792687832081974134637014026451921536576338243322267400651083805535119025415817887075652758045539565968044552126338330231434466204888993650859153585380124240540573308417330478048240203241631072371322849430883727355239704116556046700749530006852187064160849175332758172150251213637470549781080491037088372092203085237973008861896576796238915011886636658033019385943299986285181723378096425117379056207797455963451889971988904466449319007760192467211209692128894691704353648198130409333996534250878389064152054828983825841234644875996912485916827004219887033833599723481903489316488764021700996686817244736947119285629049355809027206179193628292018441744552168286541735687924729198455883895791600170372255284216915442808139396541702893732917062958054499525549626455191658842064247047426187897146172971001949767308335268505414284088528125611263685734457560292833389995980698745893243832547007166243476192958601735336260255598581701267151224204461879782815468518040925292817115377696676461775120750971210951527384637825092221708015393564320979357698186262039029460777050248162599194429941464248920952161182024344007059684970270762248243899640259750891406957836740989312390963091260380990901672119736141666856448171380781556117025832312710039041398538035351795267732240730608951176127282191528457681241590895740457571038936173983449289126574141189374690274057472401359482497502067814596008557725079835212621242944853319496441084441343866446380876967613370281793088540430288658573281302876742323336651987699532240686371751448290351615451932054274752105688318766958111191822471878078268490672607804265064578569581247796205593441336042254502454646980009177290888726099355974892680373341371214123450598691915802122913613669446370194221846225597401405625536693874723700374068542217340621938985167525416725638580266200550048001242788847319217369432802469735271132107428204697172144851088692772696511622350096452418244968500432645305761138012204888798724453956720733374364721511323104353496583927094760495687785031050687852300161433757934364774351673531859394855389527851678630166084819717354779333605871648837631164630550613112327670353108353791304451834904017538583880634608113426156225757314283269948216017294095002859515842577481167417167637534527111130468710
sTime = time.time()

def gcd(a, b):
    b = abs(b)
    while b != 0:
        r = a%b
        a,b = b,r
    return a
def gcd_mult(numbers):
    return reduce(gcd, numbers)

N = int(input())
A = [int(input()) for _ in range(N+1)][::-1]
g = gcd_mult(A)

A = [a//g for a in A]

def f(n):
    ret = A[N]
    for i in range(N-1, -1, -1):
        ret = (ret * n + A[i]) % MOD
    return ret

ans = MOD
while time.time()-sTime < 1.5:
    k = random.randrange(10**5)
    ans = gcd(ans, f(k))

def primeFactor(N):
    i = 2
    ret = {}
    n = N
    if n < 0:
        ret[-1] = 1
        n = -n
    if n == 0:
        ret[0] = 1
    d = 2
    sq = int(n ** (1/2))
    while i <= sq:
        k = 0
        while n % i == 0:
            n //= i
            k += 1
            ret[i] = k
        if k > 0:
            sq = int(n**(1/2))
        if i == 2:
            i = 3
        elif i == 3:
            i = 5
        elif d == 2:
            i += 2
            d = 4
        else:
            i += 4
            d = 2
        
    if n > 1:
        ret[n] = 1
    return ret

for i in primeFactor(ans*g):
    print(i)

Submission status: Wrong Answer

Solution code:
from itertools import cycle
from fractions import gcd
from random import randrange

# 素因数分解
def prime_factors(n):
    i = 2
    while i * i <= n:
        if n % i:
            i += 1
        else:
            n //= i
            yield i
    if n > 1:
        yield n


N = int(input())
A = list(int(input()) for _ in range(N+1))
A.reverse()


a = abs(sum(A))
b = abs(sum(a*b for a,b in zip(A, cycle((1,-1)))))
c = abs(A[0])
g = gcd(gcd(a,b),c)
P = set(prime_factors(g))

res = []

for p in P:

	C = [a%p for a in A]

	for x in range(1,p):
		def it():
			xp = 1
			for c in C:
				yield c*xp
				xp *= x
				xp %= p
		r = sum(it())%p
		if r != 0:
			break
	else:
		res.append(p)

print(*res, sep='\n')

Submission status: Wrong Answer

Solution code:
from functools import reduce
import random
import time
MOD = 59490679579998635868564022242503445659680322440679327938309703916140405638383434744833220555452130558140772894046169700747186409941317550665821619479140098743288335106272856533978719769677794737287419461899922456614256826709727133515885581878614846298251428096512495590545989536239739285290509870860732435109794045491640561011775744617218426011793283330497878491137999673760740717904979730106994180691956740996319139767941335979146924614249205862051182094081815447356600355657858936442589201739350474320966093143098768914226528498860747050474426418211313776073365498029087492308102876323562641651940555529817223923967479711293692697592800750806084802623966596666156361791014935954720420383555300213829134029560826306201326905674141090667002959274513478562033467485802590354539887133866434256562999666615074220857502563011098759348850149774083443643246907501430076141377431759984926248272553982527014011245310690986644461251576837779050124866641531488351016070300247918750613207952783437248236577750579092103591540592540838395240861443615842939673889972504776455114767522430347610500271801944701491874649306605557851914677316523034726044356368337787113190245601235705959816327016988315321850324065292964680148884818699916224411245926491625245571963741062587277736372090007125908660590314049976206281064703285728879581199596401313352724403492043120507597057004633720111008838673185885941472392623805512709896864520875740497617898566981218781313900004406341154884180472153171064617953661517880143988867189622824081729539392205701820144922307223627345876707465251206005262622236311161449785941834002795854996108322186230425179217618301526151712928790928933798369678844576216735378555233905935973195721247604933753363412045618703247367192610615234835041956551569232557323060407648325565048712478527583315981204846341857095134567470182330491262285172727037299211391244340592936174221176781260586188162350081192408213470101717320475998863222409777310443027421196981193126541663212124245716187453863438039402316877152286468198891603632606578778749292403571792687832081974134637014026451921536576338243322267400651083805535119025415817887075652758045539565968044552126338330231434466204888993650859153585380124240540573308417330478048240203241631072371322849430883727355239704116556046700749530006852187064160849175332758172150251213637470549781080491037088372092203085237973008861896576796238915011886636658033019385943299986285181723378096425117379056207797455963451889971988904466449319007760192467211209692128894691704353648198130409333996534250878389064152054828983825841234644875996912485916827004219887033833599723481903489316488764021700996686817244736947119285629049355809027206179193628292018441744552168286541735687924729198455883895791600170372255284216915442808139396541702893732917062958054499525549626455191658842064247047426187897146172971001949767308335268505414284088528125611263685734457560292833389995980698745893243832547007166243476192958601735336260255598581701267151224204461879782815468518040925292817115377696676461775120750971210951527384637825092221708015393564320979357698186262039029460777050248162599194429941464248920952161182024344007059684970270762248243899640259750891406957836740989312390963091260380990901672119736141666856448171380781556117025832312710039041398538035351795267732240730608951176127282191528457681241590895740457571038936173983449289126574141189374690274057472401359482497502067814596008557725079835212621242944853319496441084441343866446380876967613370281793088540430288658573281302876742323336651987699532240686371751448290351615451932054274752105688318766958111191822471878078268490672607804265064578569581247796205593441336042254502454646980009177290888726099355974892680373341371214123450598691915802122913613669446370194221846225597401405625536693874723700374068542217340621938985167525416725638580266200550048001242788847319217369432802469735271132107428204697172144851088692772696511622350096452418244968500432645305761138012204888798724453956720733374364721511323104353496583927094760495687785031050687852300161433757934364774351673531859394855389527851678630166084819717354779333605871648837631164630550613112327670353108353791304451834904017538583880634608113426156225757314283269948216017294095002859515842577481167417167637534527111130468710
sTime = time.time()
N = int(input())
A = [int(input()) for _ in range(N+1)][::-1]

def gcd(a, b):
    b = abs(b)
    while b != 0:
        r = a%b
        a,b = b,r
    return a
def f(n):
    ret = A[N]
    for i in range(N-1, -1, -1):
        ret = (ret * n + A[i]) % MOD
    return ret

ans = 0
while time.time()-sTime < 1.5:
    k = random.randrange(10**5)
    ans = gcd(ans, f(k))

def primeFactor(N):
    i = 2
    ret = {}
    n = N
    if n < 0:
        ret[-1] = 1
        n = -n
    if n == 0:
        ret[0] = 1
    d = 2
    sq = int(n ** (1/2))
    while i <= sq:
        k = 0
        while n % i == 0:
            n //= i
            k += 1
            ret[i] = k
        if k > 0:
            sq = int(n**(1/2))
        if i == 2:
            i = 3
        elif i == 3:
            i = 5
        elif d == 2:
            i += 2
            d = 4
        else:
            i += 4
            d = 2
        
    if n > 1:
        ret[n] = 1
    return ret

for i in primeFactor(ans):
    print(i)

Submission status: Wrong Answer

Solution code:
import math
import time
import random

t1 = time.time() 

#euclid
A=[2*k+1 for k in range(1,15813)]
p=[2]

while(1):
    y=A.pop(0)
    if y > math.sqrt(15813):
        break
    p.append(y)
    A=[x for x in A if x%y!=0]
p=p+A
N=int(input())
X=[]
for i in range(N+1):
    X.append(int(input()))
#cands
C=[x for x in p if X[-1]%x==0]
if len(C)==0 and sum([ X[-1]==x for x in [-1, 0, 1] ])==0 :
    C=[abs(X[-1])]
    

tmax=1
kflag=0
Q=0
while(len(C)!=0):
    Q+=1
    Cnew=[]
    z=random.randint(1,max(C))
    for c in C:
        if kflag:
            break
        S=0
        t2=time.time()
        zz=z%c
        tmp=1
        for i in range(N+1):
            tmp=(tmp*z)%c
            S +=  (X[i] * tmp)%c
            if ( t2-t1 ) > tmax*0.9:
                kflag=1
        if S%c==0:
            Cnew.append(c)
    if kflag:
        break
    else:
        C=Cnew[:]
for c in sorted(C):
    print(c)

Submission status: Wrong Answer

Solution code:
import sys
sys.setrecursionlimit(10**7)
INF = 10 ** 18
MOD = 10 ** 9 + 7
def POW(x, y):
    if y == 0:
        return 1
    elif y == 1:
        return x
    elif y % 2 == 0:
        return POW(x, y // 2) ** 2 % MOD
    else:
        return POW(x, y // 2) ** 2 * x % MOD
def mod_factorial(x, y): return x * POW(y, MOD - 2) % MOD
def LI(): return [int(x) for x in sys.stdin.readline().split()]
def LI_(): return [int(x) - 1 for x in sys.stdin.readline().split()]
def LF(): return [float(x) for x in sys.stdin.readline().split()]
def LS(): return sys.stdin.readline().split()
def II(): return int(sys.stdin.readline())
def SI(): return input()
from collections import Counter

# prime numbers
_nprime = 10 ** 5 + 1
primes = list(range(int(_nprime) + 1))
primes[1] = 0
for i in range(2, int((_nprime + 1) ** 0.5) + 1):
    if primes[i]:
        for j in range(2 * i, _nprime + 1, i):
            primes[j] = 0

def main():
    N = II()
    A = []
    for _ in range(N + 1):
        A.append(abs(II()))
    A = A[::-1]
    # get primes of A[0]
    p0 = set()
    a0 = A[0]
    for i in range(2, int(A[0] ** 0.5) + 1):
        if a0 % i == 0:
            p0.add(i)
            while a0 % i == 0:
                a0 //= i
    if a0 > 1:
        p0.add(a0)
    # end get primes of A[0]
    ns = []
    for i in range(10):
        n = 0
        for k in range(N + 1):
            n += A[k] * (i ** k)
        ns.append(n)
    for p in p0:
        ok = True
        for n in ns:
            if n % p != 0:
                ok = False
                break
        if ok:
            print(p)
    return 0

main()

Submission status: Wrong Answer

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LS()
    return l
sys.setrecursionlimit(1000000)
mod = 998244353

#C
def C():
    n = I()
    s = S()
    ans = float("inf")
    b = [0 for i in range(n)]
    for i in range(n):
        if s[i] == "#":
            b[i] = 1
    w = [0 for i in range(n)]
    for i in range(n):
        if s[i] == ".":
            w[i] = 1
    b.insert(0,0)
    w.insert(0,0)
    for i in range(n):
        b[i+1] += b[i]
        w[i+1] += w[i]
    for i in range(n+1):
        m = b[i]+w[n]-w[i]
        ans = min(ans,m)
    print(ans)
    return

#D
def D():
    n = I()
    return

#E
def E():
    def factorize(n):
        if n == 1:
            return []
        if n < 4:
            return [n]
        i = 2
        res = []
        m = n
        while i**2 <= n:
            if m%i == 0:
                while m%i == 0:
                    m //= i
                res.append(i)
            i += 1
        if m != 1:
            res.append(m)
        return res
    def f(x,m):
        res = a[-1]%m
        for i in range(n):
            res += a[i]*pow(x,n-i,m)%m
            if res >= m:
                res %= m
        return res
    n = I()
    a = IR(n+1)
    i = n
    while i >= 0:
        if a[i] == 0:
            a.pop(i)
            n -= 1
        i -= 1
    if n == -1:
        quit()
    ans = factorize(abs(a[-1]))
    j = 0
    while j < len(ans):
        for i in range(-50,51):
            if f(i,ans[j]) != 0:
                ans.pop(j)
                j -= 1
                break
        j += 1
    for i in ans:
        print(i)
    return

#F
def F():
    return


#Solve
if __name__ == "__main__":
    E()

Submission status: Wrong Answer

Solution code:
from collections import defaultdict

n = int(input())
A = [int(input()) for i in range(n + 1)]

def factorize(n):
    i = 2
    A = defaultdict(int)
    
    while n % i == 0:
        n //= i
        A[i] += 1
    i += 1

    while n != 1 and i * i <= n:
        while n % i == 0:
            n //= i
            A[i] += 1
        i += 2
    if n != 1:
        A[n] += 1

    return A

P = factorize(A[-1])

for p, v in P.items():
    S = 0
    for a in A:
        S += a % p
        S %= p
    if S % p == 0:
        print(p)
print()

Submission status: Wrong Answer

Solution code:
# coding: utf-8
# Your code here!
import sys, collections, bisect, math, itertools
from fractions import gcd 
#input = sys.stdin.readline #文字列入力のときは注意

m = int(input())
#s = input()
#a = [int(i) for i in input().split()]
a = [int(input()) for j in range(m+1)] # 縦「n」行

beki = 0
while a[-1] == 0:
    a.pop()
    beki+=1
if m != len(a)-1:
    m = len(a)
    a.append(0)


def get_sieve_of_eratosthenes(n):
    if not isinstance(n, int):
        raise TypeError('n is int type.')
    if n < 2:
        raise ValueError('n is more than 2')
    prime = []
    limit = math.sqrt(n)
    data = [i + 1 for i in range(1, n)]
    while True:
        p = data[0]
        if limit <= p:
            return prime + data
        prime.append(p)
        data = [e for e in data if e % p != 0]
allprimes = get_sieve_of_eratosthenes(10010)

primes=[]
def get_sieve_of_eratosthenes2():
    prime = []
    data = [i + 1 for i in range(1, 35000)]
    c=abs(a[m])
    while True:
        p = data[0]
        if 35000 <= p*p:
            return prime + data
        if c%p==0: primes.append(p)
        prime.append(p)
        data = [e for e in data if e % p != 0]

primes2=[]
def get_sieve_of_eratosthenes3():
    prime = []
    data = [i + 1 for i in range(1, 35000)]
    c=abs(a[0])
    while True:
        p = data[0]
        if 35000 <= p*p:
            return prime + data
        if c%p==0: primes2.append(p)
        prime.append(p)
        data = [e for e in data if e % p != 0]


get_sieve_of_eratosthenes2()
get_sieve_of_eratosthenes3()
#print(primes)
if len(primes) == 0 and abs(a[m]) != 1:
    primes.append(abs(a[m]))
if len(primes2) == 0 and abs(a[0]) != 1:
    primes2.append(abs(a[0]))
n=m
if a[m]==0:
    primes = allprimes


primes = primes + primes2
#print(a)
#print(primes)
for p in primes:
    if p > 10**4+5:
        if sum([i%p for i in a]) == 0:
            print(p)
        continue
    coeff = list(reversed(a))
    while len(coeff) > p:
#        print(coeff,p)
        t=coeff.pop()
        coeff[-p+1]+=t
            
#    print(coeff)
    if sum([i%p for i in coeff]) == 0:
        print(p)

Submission status: Wrong Answer

Solution code:
def prime_fact(n: int)->list:
    '''n の素因数を返す
    '''
    if n < 2:
        return []

    d = 2
    res = []
    while n > 1 and d * d <= n:
        if n % d == 0:
            res.append(d)
        while n % d == 0:
            n //= d
        d += 1

    if n > 1:
        # n が素数
        res.append(n)

    return res


def polynominal_divisors(N: int, A: list)->list:
    primes = prime_fact(abs(A[-1]))

    def f(x: int)->int:
        ret = A[0]
        for i in range(N):
            ret = ret * x + A[i+1]
        return ret

    res = []
    for p in primes:
        if f(p-1) % p == 0:
            res.append(p)
    return res


if __name__ == "__main__":
    N = int(input())
    A = [int(input()) for _ in range(N+1)]

    ans = polynominal_divisors(N, A)
    for a in ans:
        print(a)

Submission status: Wrong Answer

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LS()
    return l
sys.setrecursionlimit(1000000)
mod = 998244353

#C
def C():
    n = I()
    s = S()
    ans = float("inf")
    b = [0 for i in range(n)]
    for i in range(n):
        if s[i] == "#":
            b[i] = 1
    w = [0 for i in range(n)]
    for i in range(n):
        if s[i] == ".":
            w[i] = 1
    b.insert(0,0)
    w.insert(0,0)
    for i in range(n):
        b[i+1] += b[i]
        w[i+1] += w[i]
    for i in range(n+1):
        m = b[i]+w[n]-w[i]
        ans = min(ans,m)
    print(ans)
    return

#D
def D():
    n = I()
    return

#E
def E():
    def factorize(n):
        if n == 1:
            return []
        if n < 4:
            return [n]
        i = 2
        res = []
        m = n
        while i**2 <= n:
            if m%i == 0:
                while m%i == 0:
                    m //= i
                res.append(i)
            i += 1
        if m != 1:
            res.append(m)
        return res
    def f(x,m):
        res = a[-1]%m
        for i in range(n):
            res += a[i]*pow(x,n-i,m)%m
            if res >= m:
                res %= m
        return res
    n = I()
    a = IR(n+1)
    i = n
    while i >= 0:
        if a[i] == 0:
            a.pop(i)
            n -= 1
        i -= 1
    if n == -1:
        print()
        quit()
    ans = factorize(abs(a[-1]))
    j = 0
    while j < len(ans):
        for i in range(100,201):
            if f(i,ans[j]) != 0:
                ans.pop(j)
                j -= 1
                break
        j += 1
    if len(ans) == 0:
        print()
        quit()
    for i in ans:
        print(i)
    return

#F
def F():
    return


#Solve
if __name__ == "__main__":
    E()

Submission status: Wrong Answer

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LS()
    return l
sys.setrecursionlimit(1000000)
mod = 998244353

#C
def C():
    n = I()
    s = S()
    ans = float("inf")
    b = [0 for i in range(n)]
    for i in range(n):
        if s[i] == "#":
            b[i] = 1
    w = [0 for i in range(n)]
    for i in range(n):
        if s[i] == ".":
            w[i] = 1
    b.insert(0,0)
    w.insert(0,0)
    for i in range(n):
        b[i+1] += b[i]
        w[i+1] += w[i]
    for i in range(n+1):
        m = b[i]+w[n]-w[i]
        ans = min(ans,m)
    print(ans)
    return

#D
def D():
    n = I()
    return

#E
def E():
    def factorize(n):
        if n == 1:
            return []
        if n < 4:
            return [n]
        i = 2
        res = []
        m = n
        while i**2 <= n:
            if m%i == 0:
                while m%i == 0:
                    m //= i
                res.append(i)
            i += 1
        if m != 1:
            res.append(m)
        return res
    def f(x,m):
        res = a[-1]%m
        for i in range(n):
            res += a[i]*pow(x,n-i,m)%m
            if res >= m:
                res %= m
        return res
    n = I()
    a = IR(n+1)
    i = n
    while i >= 0:
        if a[i] == 0:
            a.pop(i)
            n -= 1
        i -= 1
    if n == -1:
        print()
        quit()
    ans = factorize(abs(a[-1]))
    j = 0
    while j < len(ans):
        for i in ans:
            if f(i,ans[j]) != 0:
                ans.pop(j)
                j -= 1
                break
        j += 1
    if len(ans) == 0:
        print()
        quit()
    for i in ans:
        print(i)
    return

#F
def F():
    return


#Solve
if __name__ == "__main__":
    E()