I was once in a meeting about a battery charger, which measured voltage from 0 to 15 in 255 equal steps. It had a threshold of something like 9.4 volts, and I rattled off 9.4117647, and the others wondered where I got all those digits from. 9412 works instead of 588 in the equation.
So using the generalization, there are no solutions with two digits numbers (base 10). 101 = 10² + 1² is the only factorization, and there are no integer solutions for 2A - 10 = +/-10 or +/- 1 with 0
I greatly appreciate your videos professor Penn. Can we please have the link to splitting a number into perfect squares in the video description as you said it exists in one of your older posts. Thank you
I looked at the problem generalized in a different way (but only in base ten) out of curiosity. 12^2 + 33^2 = 1233 88^2 + 33^2 = 8833 990^2 + 100^2 = 990100 9412^2 + 2353^2 = 94122353 17650^2 + 38125^2 = 1765038125 25840^2 + 43776^2 = 2584043776 I'll fix this algorithm too, if I can. It's too slow.
Wow!!! That is amazing, since I was thinking of this problem (the 4-digit one) as a candidate for math contest organized by KTU (Kaunas University of Technology, Lithuania). I settled on a similar one i.e. find all 4-digit xyzw such that xyzw = (xy + zw)^2. Maybe you could make a video of the solution to that one in a month. Our contest will take place on Jan. 25-th.
@rainerzufall42Are you saying we can think of A and B as one symbol rather than an n-digit number. Like if A was 25, would it mean to think of ‘25’ as a single character, kinda like F in hexadecimal?
@@CautionRamen A is xy and B is zw, just for naming conventions. It's either a block of digits or a single digit, but that doesn't matter for this problem: We are putting equal length strings of digits together and compare the value to the sum of both strings squared (the base doesn't change a thing either, as long there's no overflow and carry). So to make the problem as simple as possible, we can skip any considerations about multi-digit strings, we can use single-digit numbers to solve the problem. It will immedietely work for concatenated digits. In your example, it doesn't matter, if A is "15" (base 10) or "F" (hexadecimal), the problem is the same!
@rainerzufall42 So is considering x₁…x_n as one digit what you mean? Like if A = 25 (base 10) and B = 50 (base 10), does considering ‘25’ as a single digit in base 100, and ‘50’ as a single digit in base 100 equivalent?
The video explains how to generalize to other bases, so if you watched the video, you can see it's a clever way to check if we're paying attention and to catalyze views and engagement. Based on the content of the video, you would be wrong to assume the equation is in base 10.
@stickfiftyfive I did watch the video and actually didn't see much explination into generalizing to other bases even in the end. But it's still true that the thumbnail is wrong because we always assume base 10 unless told otherwise.
No. Note that he wrote at the beginning “In base, b 0 < A,B < b”. In other words, it's a different approach to the previous base 10 resolution. In this approach, you have to think that in the previous problem, base 100 would be used.
12=1²+2² (base 3)
Fun fact: 1/17 = 0.05882353 (when rounded) and 588^2 + 2353^2 = 5882353
I was once in a meeting about a battery charger, which measured voltage from 0 to 15 in 255 equal steps. It had a threshold of something like 9.4 volts, and I rattled off 9.4117647, and the others wondered where I got all those digits from. 9412 works instead of 588 in the equation.
An even more general equation here is, for 0
0.05882353 ≈ 1/17
and 5882353
= 588²+2353²
Where: 2353² is 1/17
of: 94122353
= 9412²+×2353²
I think we get only possibilities for the sum of squares decomposition because these numbers 10001 and 145 are a product of two primes
Can anyone help to share back the video to express any integer to sum of two square mentioned in video?
ua-cam.com/video/_VcEdkBZBMg/v-deo.htmlsi=w5mtpQGMW4uo5n14
1233 and 8833 new favorite numbers
So using the generalization, there are no solutions with two digits numbers (base 10).
101 = 10² + 1² is the only factorization, and there are no integer solutions for 2A - 10 = +/-10 or +/- 1 with 0
I greatly appreciate your videos professor Penn. Can we please have the link to splitting a number into perfect squares in the video description as you said it exists in one of your older posts.
Thank you
Is it just a coincidence or do the two solutions for A add up to the base every time?
not a coincidence at all: 2A - b = +-x results in A1 = b/2 + x and A2 = b/2 - x, thus A1 + A2 = b
I looked at the problem generalized in a different way (but only in base ten) out of curiosity.
12^2 + 33^2 = 1233
88^2 + 33^2 = 8833
990^2 + 100^2 = 990100
9412^2 + 2353^2 = 94122353
17650^2 + 38125^2 = 1765038125
25840^2 + 43776^2 = 2584043776
I'll fix this algorithm too, if I can. It's too slow.
Wow!!! That is amazing, since I was thinking of this problem (the 4-digit one) as a candidate for math contest organized by KTU (Kaunas University of Technology, Lithuania). I settled on a similar one i.e. find all 4-digit xyzw such that xyzw = (xy + zw)^2. Maybe you could make a video of the solution to that one in a month. Our contest will take place on Jan. 25-th.
For the base b case, why is it Ab + B, and not A(b^n) + B?
More general! A and B are not interested in the number base! The number of digits is only distracting, wlog we can consider 1-digit numbers!
@rainerzufall42Are you saying we can think of A and B as one symbol rather than an n-digit number. Like if A was 25, would it mean to think of ‘25’ as a single character, kinda like F in hexadecimal?
@@CautionRamen A is xy and B is zw, just for naming conventions. It's either a block of digits or a single digit, but that doesn't matter for this problem: We are putting equal length strings of digits together and compare the value to the sum of both strings squared (the base doesn't change a thing either, as long there's no overflow and carry). So to make the problem as simple as possible, we can skip any considerations about multi-digit strings, we can use single-digit numbers to solve the problem. It will immedietely work for concatenated digits.
In your example, it doesn't matter, if A is "15" (base 10) or "F" (hexadecimal), the problem is the same!
@rainerzufall42 So is considering x₁…x_n as one digit what you mean? Like if A = 25 (base 10) and B = 50 (base 10), does considering ‘25’ as a single digit in base 100, and ‘50’ as a single digit in base 100 equivalent?
Sorry I’m asking so many questions, I just find it confusing
So 2^2 = 0. ?????
Base 12. 25 in base 12 is 29, which is equal to 2²+5².
Great thumbnail, huh?
No the thumbnail is not true. 25 ≠ 2^2 + 5^2.
True in base 12
@TheEternalVortex42 But we always assume base 10 when not being said it's base 12 like in the thumbnail, so it's still wrong.
The video explains how to generalize to other bases, so if you watched the video, you can see it's a clever way to check if we're paying attention and to catalyze views and engagement. Based on the content of the video, you would be wrong to assume the equation is in base 10.
@stickfiftyfive I did watch the video and actually didn't see much explination into generalizing to other bases even in the end. But it's still true that the thumbnail is wrong because we always assume base 10 unless told otherwise.
Ok
10:00 b need to be b^n
No, the base in question is indeed b. In his first example, b = 100, and in the slight generalisation, b = 10ⁿ.
No. Note that he wrote at the beginning “In base, b 0 < A,B < b”. In other words, it's a different approach to the previous base 10 resolution. In this approach, you have to think that in the previous problem, base 100 would be used.
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