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I am trying to write a code in Python to do the following.
We can express the Erdős-Straus-conjecture in function of some polynomials $x(k), y(k), z(k) in mathbb{Q}[k]$ such that $frac4k = frac{1}{x(k)} + frac{1}{y(k)} + frac{1}{z(k)}$. Notice that the coefficients are rational numbers, but they have to be integers to be able to use them for this conjecture.
Following formulas are given:

1. $ frac4k = frac1{uk} + frac1{vk} + frac{4uv - u - v}{uvk} quad(P1)$


2. $ frac4k = frac1{uk} + frac{4u-1}{uk+v} + frac{(4u-1)v}{uk(uk+v)}quad (P2) $


3. $ frac4k = frac{4u}{uk+v+1} + frac4{k(uk+v+1)} + frac{4v}{k(uk+v+1)}quad (P3)$


4. $ frac4k = frac4{k+v} + frac{4v}{k(k+4u+v)} + frac{16uv}{k(k+v)(k+4u+v)} quad (P4) $

The following theorem is used to determine when polynomials yield integer values:
Consider $f in mathbb{Q}[k]$ en choose $m in mathbb{Z}$ such that $m cdot f$ only has integer coefficients. Now consider $n = am+b in mathbb{Z}$ with $0 le b < m$. Then $f(n) in mathbb{Z}$ iff $(mcdot f)(b) equiv 0,,(mod,,m)$.

This theorem implies that only looking at the residue classes modulo $m$ is sufficient to find out for which integers $n$ also $f(n)$ will be an integer.

For the Erdös-Straus-conjucture I have three polynomials $f_1(n), f_2(n), f_3(n)$. I want to find out for which $n$ all three polynomials have integer coefficients. Consider the corresponding $m$-values $m_1,m_2,m_3$ and let $m$ be equal to $lcm(m_1,m_2,m_3)$. Applying the above theorem (with the $m$ found), it's enough to check for which residue classes modulo $m$ all three polynomials $f_1,f_2,f_3$ will be integers.

So, I'd like to write a function in Python that has concrete values of $u$ and $v$ as input. I want to calculate the correct value for $m$ and to check for which residue classes modulo $m$ the polynomials have integer values. Therefore, the output should be a tuple ($m$,L) with L the list of residue classes modulo $m$ (that satisfy the conditions). I want to be able to use this function for (P1), (P2), (P3) and (P4).

This is what I've got so far:

P.<k> = PolynomialRing(QQ)

var('x,y,z')

def P1(u,v):

    x = u*k

    y = v*k

    z = (u*v*k)/(4*u*v-u-v)

    return (x,y,z)

def P2(u,v):

    x = u*k

    y = (u*k+v)/(4*u-1)

    z = (u*k*(u*k+v))/((4*u-1)*v)

    return (x,y,z)

def P3(u,v):

    x = (u*k+v+1)/(4*u)

    y = (k*(u*k+v+1))/4

    z = (k*(u*k+v+1))/(4*v)

    return (x,y,z)

def P4(u,v):

    x = (k+v)/4

    y = (k*(k+4*u+v))/(4*v)

    z = (k*(k+v)*(k+4*u+v))/(16*u*v)

    return (x,y,z)



var('m,k')

def findM(f):

    m = (f(k=1)^-1).denominator() 

    return m  

var('u,v,P,m1,m2,m3,m,k')

def mP(u,v,P):

    D = P(u,v)

    m1 = findM(D[0])

    m2 = findM(D[1])

    m3 = findM(D[2])

    m = lcm([m1,m2,m3])

    L = 

    for i in xrange(1,m-1):

         if((D[0](k=i)^-1).denominator() == 1 and (D[1](k=i)^-1).denominator() == 1 and (D[2](k=i)^-1).denominator() == 1 and Mod(m*((D[0]^-1) + (D[1]^-1) + (D[2]^-1))(k=b),m) == 0):

             L.append(i)

    return (m,L)

The problem is a keep getting an empty list. I'm using Sage Math to compute this (which is based on Python - kinda).
Can someone please help me.

Infinitely grateful!

The following is a possible implementation, which you can test here.

R. = PolynomialRing(QQ)

D. = PolynomialRing(ZZ)

def P1(u,v):

    x = u*k

    y = v*k

    z = (u*v*k)/(4*u*v-u-v)

    return (x,y,z)

def P2(u,v):

    x = u*k

    y = (u*k+v)/(4*u-1)

    z = (u*k*(u*k+v))/((4*u-1)*v)

    return (x,y,z)

def P3(u,v):

    x = (u*k+v+1)/(4*u)

    y = (k*(u*k+v+1))/4

    z = (k*(u*k+v+1))/(4*v)

    return (x,y,z)

def P4(u,v):

    x = (k+v)/4

    y = (k*(k+4*u+v))/(4*v)

    z = (k*(k+v)*(k+4*u+v))/(16*u*v)

    return (x,y,z)

def findM(f):

    m = f.denominator() 

    return m

def mP(u,v,P):

    f = P(u,v)

    m1 = findM(f[0])

    m2 = findM(f[1])

    m3 = findM(f[2])

    m = lcm([m1,m2,m3])

    F = [D(m*f[0]),D(m*f[1]),D(m*f[2])]

    L = 

    for i in xrange(1,m+1):

         if(Mod(F[0](i),m)==0 and Mod(F[1](i),m)==0 and Mod(F[2](i),m)==0):

             L.append(i)

    return (m,L)

#Example:

[u,v,P] = [3/5,5,P4]

[m,S] = mP(u,v,P)

[x,y,z] = P(u,v)



print("[u,v,m] = ",[u,v,m])

print("[k,x(k),y(k),z(k),check]:")

for k in S:

    [xk,yk,zk] = [x(k),y(k),z(k)]

    chk = 1/xk + 1/yk + 1/zk

    print [k,xk,yk,zk,chk==4/k]

The example, using $(u,v)=(3/5,5)$ and $P4$, will return

[75, 20, 309, 10300, True]

[175, 45, 1596, 119700, True]

[375, 95, 7170, 1135250, True]

[475, 120, 11457, 2291400, True]

[675, 170, 23031, 6525450, True]

[775, 195, 30318, 9853350, True]

[975, 245, 47892, 19555900, True]

[1075, 270, 58179, 26180550, True]

The first line would mean
$$
frac{4}{75} = frac{1}{20} + frac{1}{309} + frac{1}{10300}
$$

You declared

P.&ltk&gt = PolynomialRing(QQ)

but later this removes the declaration:

var('u,v,P,m1,m2,m3,m,k')

You might want to assign a universal ring $R:=mathbb Q[k]$ that is untouched throughout instead. Might also be a good idea to have an integral ring $D:=mathbb Z[k]$ since you will work in it later, but I guess it's not necessary. i.e.

R.&ltk&gt = PolynomialRing(QQ)  

D.&ltk&gt = PolynomialRing(ZZ)

For finding $m$, it suffices to use

def findM(f):

    m = f.denominator() 

    return m  

The current command

(f(k=1)^-1).denominator()

will find the numerator instead, and evaluated at $f(1)$ so not the true denominator.

One last point is xrange(1,m-1) gives you the range $[1,2,dots,m-2]$ but you want $[1,2,dots,m]$.