I started with the obvious try-out: one of them is zero and found two solutions (0,1976) and (1976,0). The other obvious one to try is x=y yielding the (494,494). I only did not know I had them all.
y=1976+x-2*sqrt(1976*x) has infinite integer solutions, but only three are valid for the given equation. This was a nice problem because it made me think. Making us think is perhaps the best reason to do math.
Hi. For a while now, the sound and image (of your last videos) are not synchronized. We can hear the sound of your pen and a while after we can see what you are writing. It's a bit disturbing. Can you check if it comes from your "side" or may be i'm the only one to have that issue. Thx
@WahranRai Complex numbers are any numbers c = a+bi with a and b being any kind of real numbers. And I'm not including those here. What you call intergers, and often is called so, are the real integers. And they need to be called real integers, if you want to make sure to exclude the complex integers, the so-called Gaussian integers. Else the term 'integers' becomes inconsistent.
@@pepebriguglio6125 An integer is a whole number (not a fractional number) that can be positive, negative, or zero. The set of integers, denoted Z, is formally defined as follows: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Maybe Wolfram missed the other two solutions because of computer rounding error, since they are all irrationals.
I started with the obvious try-out: one of them is zero and found two solutions (0,1976) and (1976,0). The other obvious one to try is x=y yielding the (494,494). I only did not know I had them all.
More easy If You note :x=496axa and y=496bxb;a+b=2, a,b are integers!
y=1976+x-2*sqrt(1976*x) has infinite integer solutions, but only three are valid for the given equation. This was a nice problem because it made me think. Making us think is perhaps the best reason to do math.
Absolutely!
we have y=2×13×19×t^2
wolfram alpha says "Standard computation time exceeded... Try again with Pro Computation Time" 😀
hehe trying to sell you pro stuff
247=13*19
I used the second method after finding the prime factorization for 1,976.
Hi. For a while now, the sound and image (of your last videos) are not synchronized. We can hear the sound of your pen and a while after we can see what you are writing. It's a bit disturbing. Can you check if it comes from your "side" or may be i'm the only one to have that issue. Thx
video is lagging
Issue with screen recording. I don't know why that's happening but restarting the iPad fixed it for now.
Couldn't Couldn't y be 1÷1976 though?
integer?
@@SyberMath He would have a hard time listing them if they were real !
x = y = 494
1976 == 4 X 494. So sqrt (1976) == 2 X sqrt(494). Thus x == y == 494 is one solution. Now onto the next video !!!
√(x) + √(y) = √(1976)
√(x) + √(y) = √(2³×13×19)
√(x) + √(y) = 2√(494)
Integer solutions:
=> x=494a² and y=494b²
√(494a²) + √(494b²) = 2√(494)
a√(494) + b√(494) = 2√(494)
=> a+b = 2
Real solutions
=> min(x,y) = 0
=> max(x,y) = 1976 = 4×494
=> max(a²,b²) = 4
=> max(a,b) = 2
=> (a,b) € {(2,0),(0,2),(1,1)}
=> (x,y) € {(1976,0),(0,1976),(494,494)}
Complex solutions:
a = i => b = 2-i
(a,b) = (i,2-i)
(a²,b²) = (-1,3-4i)
(x,y) = (-494,1482-1976i)
Etc.
He stated that it is diophantine equation meaning solutions are integers.
Why complex solutions ?
@WahranRai
Complex numbers are any numbers c = a+bi with a and b being any kind of real numbers. And I'm not including those here. What you call intergers, and often is called so, are the real integers. And they need to be called real integers, if you want to make sure to exclude the complex integers, the so-called Gaussian integers. Else the term 'integers' becomes inconsistent.
@@pepebriguglio6125 An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
The set of integers, denoted Z, is formally defined as follows:
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
@@WahranRai
I know. Read again please.
@@pepebriguglio6125 You know NOTHING, stop butchering maths
1976 2
988 2
494 2
247 13
19 19
So it makes sense that X and Y