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9

I'll show the resultant formulation for the degree 15 example. The polynomial in question: poly15 = (-282300416 - 64550400 # - 34426880 #^2 - 14185880 #^3 + 8564800 #^4 + 4231216 #^5 - 972800 #^6 - 367820 #^7 + 27360 #^8 + 2600 #^9 + 1680 #^10 + 100 #^11 - 240 #^12 + 40 #^13 + #^15) &[z]; Define monic polynomials of degree 3 and 5 ...

12

A constructive approach The problem can be solved if the form of the solution is given. Define the two factors using a hint (that these should be cubic equations) in the original post y1 = x^3 - p1 x + q1 y2 = x^3 - p2 x + q2 Build a companion matrix of the polynomial $p(x)$ CompanionMatrix[p_,x_]:=Module[{n,w=CoefficientList[p,x]},w=-w/Last[w]; ...

6

Your G is called IntegerExponent[] in Mathematica: Table[IntegerExponent[5^2 7^3 11^4, k], {k, {5, 7, 11}}] {2, 3, 4} You should now be able to use that function to write your function F.

1

(This is a self-answer after discovering I can use Reduce directly.) The previous answer gives ideas related to your approach of iterating palindromes. Here, I present a way to directly compute them instead. Directly computing $2,3$-palindromes "...whether there is a way to do this computation faster?" Rather than brute-force checking palindromes, you ...

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