@LucMeekes wrote:
Mathcad /Maple will not be fooled:
If you evaluate that numerically, it says 2048,
but that's due to poor (with respect to symbolic) numerical precision,
similar to the numerical results I presented above.
Luc
Yes, I'd expect that. However, I do note that Tetsuro specifically mentioned Prime 7.
If, as I implied by my mention of infinities, you plug in -∞ to the expression in Maple you get 2048 symbolically (at least, in Maple Flow you do).
(Mathematica gives "Sum[2^n, {n, -Infinity, 10}] = 2048", as well. I would hope Mathcad Prime 7 gives the same result?).
Stuart
(Of course, there's the thorny issue of whether -Infinity counts as an integer)
@ttokoro wrote:
2^11 is 2048. How to find a and b? a and b are integers.
AFAIA, the binary expansion of any integer is unique.
Stuart
Prime 7 shows another a and b.
@ttokoro wrote:
Prime 7 shows another a and b.
Given the Puzzles and Games Label, I thought it might. 😈
Stuart
Can you upload your worksheet?
@VladimirN wrote:
Can you upload your worksheet?
Хорошо. I've modified it slightly to make it more self-contained.
Stuart
Edited to add: Why is it I only notice mistakes *after* I've posted something? 🙂
The original image had the type of the function num2base as returning a String; this was because I copied the function from another worksheet where it made more sense to return a string and I forgot to change the type when I modified the function. Ridiculum est me!
Hmmm,
Never expected this would happen.
But then, if b is a positive integer <10 then a is complex
for b=10 I get an exception due to trying to take ln(0).
for b=11 , a=11.
for b>11 a is a positive real.
Ah! b=22 gives a=23 (numerically), but:
And with every higher multiple of 11 for b you get closer to a being b+1. for b=1012:
and you don't even need multiples of 11 anymore. For b=1013 the evaluation gives:
But at b=1023 the numerical evaluation fails.
Luc
Neat.
Something else that occurred to me in my sleep-deprived state early this morning, but which I promptly forgot about until now, was based on one of the many peculiarities of infinities.
(a = -38 does the trick numerically, but I wanted to give the impression of a really large number)
I don't have a symbolic processor to hand to check its result ...
Stuart
Mathcad /Maple will not be fooled:
If you evaluate that numerically, it says 2048,
but that's due to poor (with respect to symbolic) numerical precision,
similar to the numerical results I presented above.
Luc
@LucMeekes wrote:
Mathcad /Maple will not be fooled:
If you evaluate that numerically, it says 2048,
but that's due to poor (with respect to symbolic) numerical precision,
similar to the numerical results I presented above.
Luc
Yes, I'd expect that. However, I do note that Tetsuro specifically mentioned Prime 7.
If, as I implied by my mention of infinities, you plug in -∞ to the expression in Maple you get 2048 symbolically (at least, in Maple Flow you do).
(Mathematica gives "Sum[2^n, {n, -Infinity, 10}] = 2048", as well. I would hope Mathcad Prime 7 gives the same result?).
Stuart
(Of course, there's the thorny issue of whether -Infinity counts as an integer)
Yep, here you are:
I'm afraid infinity, nor its counterpart -infinity counts as an integer. Small test:
If its equal to itself plus 0.5, then it cannot be an integer:
Luc
Many thanks to all mathcsd masters.
My another answer is Stuart's one.
Fourier series of non-sinusoidal wave forms sometimes have infinity Fourier series.
Such as rectangular wave of magnitude 1 shows it as follows.
The RMS value is 1. And Mathcad can show it.
And in this case Stuart shows
This is 1111111111.11111111111111111...
> a and b are integers
Mathcad never was able to solve diophantic equations and the symbolics ever so often ignore an "assume, xxx=integer"
You probably are aware thats its quite easy to manually arrive at the unique solution a=b=11.
Given that the sum is
with a<=b<=11, you can do as follows
Somewhat shorter:
But the shortest sure is Stuarts remark about the binary representation of a number being unique.