Pol4Pal Calculator

MENUToday we serve quadratic polynomials

Enter a numerical palindrome (sequence of positive natural numbers separated by spaces)

Receive the polynomial that gives all possible n ∈ N such that the continued fraction of n starts with [z, «palindrome», 2z] for some z

«Your polynomial will display here»

How to use this calculator

  1. Choose a palindrome (ex.: "3 4 1 4 3" or "2 13 13 2")
  2. If it has even length
    1. Enter the first half (2 13 13 2)
    2. Leave "Middle element" blank
  3. If it has odd length
    1. Enter the elements before the middle (3 4 1 4 3)
    2. Enter the middle element (3 4 1 4 3)
  4. Click Calculate to get the polynomial

How polynomials are calculated

Palindrome to polynomial mapping:

Palindrome Polynomial
[   ] (Empty) f(x) = x2 + 1 x ∈ N
[2m] f(x) = (m2x + 1)x x > 0
[2m − 1] f(x) = ((2m − 1)2x + 2) x > 0
[1, 2m − 1, 1] f(x) = ((2m + 1)2x + (2m + 1)2 − 2)(x + 1) x ∈ N

For the other cases, read the paper

Help

Credits

Mathematics
Andrzej Solecki

Informatics
Igor da Silva Solecki

Paper
Code (Python)