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Golden polynomials (Posted on 2025-04-21)

Let φ be the positive root to x² = x + 1. In how many ways can φ⁶ be expressed as a degree 4 polynomial in φ with nonnegative integer coefficients? For example, one way φ⁶ can be written as a degree 5 polynomial in φ with nonnegative integer coefficients is φ⁶ = φ⁵ + φ⁴.

No Solution Yet Submitted by Danish Ahmed Khan

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computer solution

Comment 1 of 1

There are 12 ways.

phi=(1+sqrt(5))/2

ct=0;

for a=1:3

for b=0:5

for c=0:7

for d=0:12

for e=0:18

t=a*phi^4+b*phi^3+c*phi^2+d*phi+e ;

if abs(t-phi^6)<.0000001

disp([a b c d e])

ct=ct+1;

end

   coefficients of powers of phi

pwr: 4     3     2     1     0

     1     0     0     5     3
     1     0     1     4     2
     1     0     2     3     1
     1     0     3     2     0
     1     1     0     3     2
     1     1     1     2     1
     1     1     2     1     0
     1     2     0     1     1
     1     2     1     0     0
     2     0     0     2     1
     2     0     1     1     0
     2     1     0     0     0
ct =
    12

Posted by Charlie on 2025-04-21 10:23:21

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