Maxima fails to recognize some expressions as equal

[sagetrac-aginiewicz]

Maxima fails to regard some expressions as equal:

sage: value_1 = 1-golden_ratio
sage: value_2 = -golden_ratio^(-1)
sage: bool(value_1 == value_2)
True
sage: bool(value_1^x != value_2^x)
True

while

sage: bool(((x+1)^2)^y == (x^2+2*x+1)^y)
True
sage: sin(0,hold=True)^x == 0^x
sin(0)^x == 0^x
sage: bool(sin(0,hold=True)^x == 0^x)
True

Previous description:

I tried to define Fibonacci sequence using golden ratio in two ways, using values:

sage: value_1 = 1-golden_ratio
sage: value_2 = -golden_ratio^(-1)
sage: bool(value_1 == value_2)
true

(gives true, so two definitions, F1 and F2 below should be equal, even though they are not according to Sage)

sage: F1(k) = (golden_ratio^k-(value_1)^(k))/sqrt(5)
sage: F2(k) = (golden_ratio^k-(value_2)^(k))/sqrt(5)
sage: bool(F1(k) != F2(k))
true

When simplified everything seems to be equal at least for first 10 or 1000 elements:

sage: [(F1(j)-F2(j)).full_simplify() for j in range(10)]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Anyway, now to the error: limit for F1 gives wrong result:

sage: limit(F1(k+1)/F1(k), k=oo)
0

and for F2 works OK:

sage: limit(F2(k+1)/F2(k), k=oo)
1/2*sqrt(5) + 1/2

I've tested it with Sage 5.12 and 5.11, with same result. This can be as simple as some thing with how golden ratio is handled, or something far more involved maybe?

CC: @kcrisman @sagetrac-jakobkroeker

Component: calculus

Keywords: limit, golden_ratio

Stopgaps: wrongAnswerMarker

Issue created by migration from https://trac.sagemath.org/ticket/15395

[sagetrac-aginiewicz]

comment:5

I don't know when it happened, but in 6.4.1 the limit works:

sage: limit(F1(k+1)/F1(k), k=oo)
1/2*sqrt(5) + 1/2

but the comparison

sage: F1(k) = (golden_ratio^k-(value_1)^(k))/sqrt(5)
sage: F2(k) = (golden_ratio^k-(value_2)^(k))/sqrt(5)
sage: bool(F1(k) != F2(k))
true

is still wrong. Slightly minified example:

sage: value_1 = 1-golden_ratio
sage: value_2 = -golden_ratio^(-1)
sage: bool(value_1 == value_2)
True
sage: bool(value_1^x != value_2^x)
True

[sagetrac-jakobkroeker]

Stopgaps: wrongAnswerMarker

[mkoeppe]

comment:8

Unchanged after #30063 Maxima 5.44.0

[kcrisman]

comment:9

Thanks very much for checking up on all these old tickets.

[mkoeppe]

comment:11

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review.