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(This question is the 'sequel' to my first question, that you can find here: Erdős-Straus-conjecture using polynomials in Python).

With the help of contributor Yong Hao Ng, I managed to write a Sage function to calculate the value of an integer $m$ such that the coefficients in $frac4k = frac1{x(k)} + frac1{y(k)} + frac1{z(k)}$ (1) would be integers too. One thing I noticed was that $x(k),y(k)$ and $z(k)$ represent integers when $k equiv b$ (mod $m$) $iff k = am + b$ ($a,b,m in mathbb{Z}$) (Am I correct?). This means that (1) can be rewritten as follows: $frac4{am+b} = frac1{x(am+b)} + frac1{y(am+b)} + frac1{z(am+b)} = frac1{x'(a)} + frac1{y'(a)} = frac1{z'(a)}$, with $x'(a),y'(a)$ and $z'(a)$ polynomials in $a$ that are automatically integers for all $a in mathbb{Z}$.

I have done this substitution for the four polynomials that I mentioned in my previous question.

m = mP(u,v,P1)[0]

show(p1(k=a*m+b))

m = mP(u,v,P2)[0]

show(p2(k=a*m+b))

m = mP(u,v,P3)[0]

show(p3(k=a*m+b))

m = mP(u,v,P4)[0]

show(p4(k=a*m+b))

I have checked the output and I seems correct. (For example for (P1) I get back: $frac1{(160a+b)u} + frac1{(160a+b)v} + frac{4uv-u-v}{(160a+b)uv}$).

So, now to the question. As you can see, this method (of substitution) is very useful for a lot of residue classes (especially for all residue classes up to modulo 840, except for six of them!). I've done my research and found that the six remaining open residue classes should be $1^2, 11^2, 13^2, 17^2, 19^2$ and $23^2$ mod($m$). Alas, I can't find a way to write a useful Sage (or Python) function to find these classes. Of course I've already tried something:

resid = 

for i in xrange(1,841):

    resid.append(i)



var('u,v')

L = 

for u in [1..8]:   %I found that looking for values of u and v between 1 and 8 should be enough, to find the six needed residue classes.

    for v in [1..8]:



    mP1_ = mP(u,v,P1)

    if(mP1_[0] in divisors(840)):

        L.append(mP1_[0])

        for i in range(mP1_[0]):

            if(i not in L):

                L.append(i)



    mP2_ = mP(u,v,P2)

    if(mP2_[0] in divisors(840)):

        L.append(mP2_[0])

        for i in range(mP2_[0]):

            if(i not in L):

                L.append(i)



    mP3_ = mP(u,v,P3)

    if(mP3_[0] in divisors(840)):

        L.append(mP3_[0])

        for i in range(mP3_[0]):

            if(i not in L):

                L.append(i)



    mP4_ = mP(u,v,P4)

    if(mP4_[0] in divisors(840)):

        L.append(mP4_[0])

        for i in range(mP4_[0]):

            if(i not in L):

                L.append(i)

Note: the code for the function mP(u,v,P) can be found in my last post.

The list I get back only consists of elements from 0 to 140 (some repeated) and the values $1^2, 11^2$ do appear in it.

I have been searching for days now and I'm really struggling. Can someone help me please. (Wow, this post was huge. Thank you for coming to my TED-Talk).

(If there is any code missing, leave a comment down below and I'll add it. Thank you.)

Immeasurably grateful!