An interesting equation

Eulero · 2016-09-15 02:29:47

Hi,
Today I want to propose you a beautiful problem:
Solve the equation:
mant{x^(-1)}=mant{x}=mant{x^2}
Where mant{x} is the mantissa function

phrontister · 2016-09-15 03:15:41

Hi Eulero;

Eulero · 2016-09-15 03:16:32

It's a solution but not the only one

bobbym · 2016-09-15 06:59:46

Hi;

Eulero · 2016-09-15 07:44:26

Good job!
But the fourth solution you proposed is wrong as you can see here. The exercise exalts one of the most curious proprieties of golden ratio : it's the only not integer number whose himself and reciprocal and square have the same fractionary part. When you have time would you post the execution?

Last edited by Eulero (2016-09-15 07:46:15)

bobbym · 2016-09-15 09:05:02

Hi;

From the link you gave, mant is the positive fractional part .

So:

f[x_] := Abs[FractionalPart[x]];
n = -(8/5); 
FindRoot[f[1/x] == f[x] && f[1/y] == f[y^2] && f[z] == f[z^2], {{x, n}, {y, n}, {z, n}}, WorkingPrecision -> 50]

yields a root at

{x -> -1.6180339887498948482045868343656381177203091798058,
y -> -1.6180339887498948482045868343656381177203091798058,
z -> -1.6180339887498948482045868343656381177203091798058}

which is

When you have time would you post the execution?

My solution is a numerical one and I will post it when I can.

zetafunc · 2016-09-15 18:54:02

Last edited by zetafunc (2016-09-16 01:52:00)

thickhead · 2016-09-15 20:57:02

When x=-1.618033989
floor(x)= -2
mant(x)=-1.618033989-(-2)=0.381966011
So

does not form a solution.

bobbym · 2016-09-16 01:29:51

Are you guys using the absolute value as the page describes?

zetafunc · 2016-09-16 01:35:00

I interpreted "positive fractional part" as being (no absolute values). I guess "negative fractional part" would be I think the error could be coming from the fact that you defined as when perhaps it should not have the absolute values (unless I have misunderstood the OP's definition of mant(x) from the Wolfram article).

Last edited by zetafunc (2016-09-16 01:35:22)

bobbym · 2016-09-16 01:36:55

Hi;

The page he gave http://mathworld.wolfram.com/Mantissa.html describes it like that.

zetafunc · 2016-09-16 01:43:15

Are you sure? I am interpreting it as how they've written it down, i.e. so that, for instance, the positive fractional part of 5.2 is 0.2, whilst its negative fractional part is 0.8. Their description does not include the word "of" between "positive fractional part" and the formula they give -- in other words, I think they're just calling that formula "positive fractional part", rather than saying to take the absolute value of that formula. That formula is how the mantissa is usually defined (although in for example analytic number theory nobody uses the word "mantissa", they just write the formula out).

Last edited by zetafunc (2016-09-16 01:44:43)

bobbym · 2016-09-16 01:46:53

Hi;

Nope, I am not sure. I will agree then that - phi is not a solution.

Eulero · 2016-09-16 03:19:46

Good job zetafunc!

thickhead · 2016-09-16 03:43:12

Hi Bobbym,
If you recall characteristic and mantissa of log to the base 10 in logarithm tables, the concept becomes clear.

bobbym · 2016-09-16 04:06:33

Hi;

If you recall characteristic and mantissa of log to the base 10 in logarithm tables, the concept becomes clear.

I do not agree. Eric's page is ambiguous to me in this case.

I agree with Knuth:

Knuth wrote:

For instance Knuth adopts the third representation 0.12345 × 10+3 in the example above, and calls 0.12345 the fraction part of the number; he adds:[7] "[...] it is an abuse of terminology to call the fraction part a mantissa, since this concept has quite a different meaning in connection with logarithms [...]".

But as this is the interpretation that the OP wanted I can only blame myself for not asking him to clear up the confusion in how I see that page.

zetafunc · 2016-09-16 04:29:18

It might have been confusing because the Wolfram page says the phrase "positive fractional part" immediately before the formula, which is somewhat tautological (if the author had just written the formula, that would have been made clearer to the reader). But the lack of the word "of" between that phrase and the formula makes it technically unambiguous, I think. (But easy to misinterpret.)

It is made slightly worse by the fact that its counterpart "negative fractional part" is never explicitly defined on the website -- at least, I cannot seem to find it. But I'm assuming that if it does exist, it would have a definition something like the one I gave in post #10.

Last edited by zetafunc (2016-09-16 04:34:23)

bobbym · 2016-09-16 08:46:16

Yes, but even the alphabet is confusing to a bumpkin.

Math Is Fun Forum

#1 2016-09-15 02:29:47

An interesting equation

#2 2016-09-15 03:15:41

Re: An interesting equation

#3 2016-09-15 03:16:32

Re: An interesting equation

#4 2016-09-15 06:59:46

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#5 2016-09-15 07:44:26

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#6 2016-09-15 09:05:02

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