WOLFRAM|DEMONSTRATIONS PROJECT

Golden Integers

​
p
11
19
29
31
real
n = m
base p
m
50
n
50
Given: solutions of
2
x
-x -1 ≡ 0 (mod
m
p
)
x
1
= 312688723022696024345230675062900261450540536087727822444008802560403034464
x
2
= 57211583938064033911824447729982475848051753436026455583514967925838269538
Check 1:
2
x
(mod
n
p
),
expecting x + 1
2
x
1
≡
312688723022696024345230675062900261450540536087727822444008802560403034465
2
x
2
≡
57211583938064033911824447729982475848051753436026455583514967925838269539
Check 2: x(x - 1) (mod
n
p
),
expecting +1
x
1
(
x
1
- 1) ≡
1
x
2
(
x
2
- 1) ≡
1
What about?
x
1
x
2
(mod
n
p
),
expecting -1
x
1
x
2
≡
369900306960760058257055122792882737298592289523754278027523770486241304000
2
x
1
2
x
2
≡
1
The positive root of the golden polynomial
2
x
-x-1
is the famous golden ratio
x
1
=ϕ≈1.61803
and the negative root is
x
2
=-
1
ϕ
≈-0.61803
. There are two integer solutions
x
1
and
x
2
to the congruence
2
x
-x-1≡0(mod
m
p
)
, where
p
is a prime and
m
is a positive integer (in this Demonstration,
p=11,19,29,31
and
m=1,2,…,50
). Can either
x
1
or
x
2
be called a "golden integer" that is analogous to
ϕ
?