So the key to such a problem "working" is that the RHS has to be the SQRT of a integer in the form K x L^2 (in the present case, K = 7, L = 17). Then m + n = L, and there will be L + 1 solution pairs (a, b) = (K x m^2, K x n^2), with m = 0, 1, 2....L and n = L, L-1, L-2....0. Another example would be SQRT(a) + SQRT(b) = SQRT(18). Here K = 2, L = 3, so m + n = L = 3 and the L + 1 = 4 solutions are (a, b) = (0,18), (2,8), (8,2), (18,0).

Wow so much nicely explained! Great technique. SO much impressed by the clean writing with different colours!!! Well done. Such a mesmerizing channel. #SUPERMATHS

I'll give some correction only. In this case m and n are integer too. Because quantity 0 is not natural, but at the same time we admit to m or n to be equal 0. For example, for pairs either m=0; n =17 or m=17; n=0. Repeat again: zero is not natural! It's important to know.

I think m times n can be derived by 1st equation and it is able to limit series of m,n. More thinking is needed to solve this problem perfectly

nice, its a much better solution than I found.

Thank you teacher for teaching lesson is the best way keep going dear teacher i say always happy day

Very interesting! 😇👏👏👏

Very nice!

Has it been specified that a and b are integer in the text? Otherwise it is a false assumption.. Ups my bad, I have not seen the first line

@The_fog_general

Жыл бұрын

Thank you , it didn’t make sense unless we know this assumption

I can find the results by logic and mental calculation but i can not make a beautiful explanation like that!

As you said , m and n belong to set of natural numbers, then how can (0,17) be solution.?

@shubhankardatta2437

Жыл бұрын

At some places around the world 0 is also considered a natural number.

È radicalizzabile la numerica dall ' alfa ?

Beautiful

Nice

Greetings from the BIG SKY. Didn't look easy to these old eyes.

Спасибо

This is easy here is a better explanation to you: To solve the equation √a + √b = √2023 for the variables a and b, we need to isolate one of the variables on one side of the equation. First, we can square both sides of the equation to eliminate the square roots: (√a + √b)^2 = (√2023)^2 Simplifying the left-hand side using the distributive property, we get: a + 2√ab + b = 2023 Next, we can move the square root term to the left-hand side and square both sides again to eliminate the remaining square root: a + b - 2023 = -2√ab Squaring both sides gives: (a + b - 2023)^2 = 4ab Expanding the left-hand side using the distributive property, we get: a^2 + b^2 + 2023^2 + 2ab - 2a*2023 - 2b*2023 = 4ab Simplifying and rearranging terms, we get: a^2 - 2ab + b^2 = 2023^2 - 4ab Now we have an equation relating a and

American School kids: "Its just sqrt2020 + sqrt3 duh"

Easy. a=0 b=2023 or a=2023 b=0 (And a lot of answers...)

@JPTaquari

Жыл бұрын

I agree whith You!

Isn't it (2*sqrt(2023*b)) that's a natural number?

@RoelPompen

Жыл бұрын

The missing step is the insight that this can be rewritten as: 2023-a+b=2*sqrt(2023b), which tells us lhs is even. From there you can assume sqrt(2023b) to be integer.

@MichaelJamesActually

Жыл бұрын

@@RoelPompen thank you! I also realized that b is an integer by definition. Not sure why that slipped past me

Très bien sauf l'écriture de votre ensemble de définition ; tel que vous l'avez écrit a et b pourraient être tous deux négatifs

Are you by any chance Jia Lou? Dr.Lou?

@mathwindow

Жыл бұрын

Sorry, I'm not

Explain box should be not arquipy the written explaination. It causes irritation

How do you know that 7b is a square number?

@mathwindow

Жыл бұрын

since √7b is an integer

@ganzir9697

Жыл бұрын

@@mathwindow Why does that imply that 7b is a square number?

@ganzir9697

Жыл бұрын

It just says that 17 * sqrt(7b) is a natural number. It doesn't say that sqrt(7b) is a natural number. That would only be the case if the natural number on the right-hand side is divisible by 17.

I thought it was sqrt(2023)/2

a=343, b=700.

A= 2023 b=0😅

Me: solve the equation sqrt(a)+sqrt(b)=sqrt(2023) ChatGPT: This is a Diophantine equation, which is an equation that seeks solutions in integers. To solve the equation sqrt(a) + sqrt(b) = sqrt(2023), we can square both sides to get rid of the square roots: a + 2*sqrt(a)*sqrt(b) + b = 2023 Then we can rearrange the equation to get: a + b = 2023 - 2*sqrt(a)*sqrt(b) This equation can be solved by guessing and checking values of a and b that satisfy it. One solution is a = 961 and b = 1062. The other solutions are found by taking the same values and switching them around. a = 1062 and b = 961 It is important to note that this is not the only solution and there are infinite solutions to this equation.

@fernandosantos3576

Жыл бұрын

from math import sqrt def find_roots(): for a in range(1, 2024): for b in range(1, 2024): if sqrt(a) + sqrt(b) == sqrt(2023): print(f"a = {a:4}, b = {b:4}") find_roots() a = 28, b = 1575 a = 112, b = 1183 a = 175, b = 1008 a = 252, b = 847 a = 343, b = 700 a = 448, b = 567 a = 567, b = 448 a = 700, b = 343 a = 847, b = 252 a = 1008, b = 175 a = 1183, b = 112 a = 1575, b = 28

Very interesting problem, thank you.😁 May I suggest a simple numerical solution for a and b as positive integers: note 2023=17^2 * 7 --> sqrt(2023)=17 * sqrt(7). Divide both sides by sqrt(7) (1) sqrt(a/7) + sqrt(b/7) = 17 Let 17 = x + y {1+16, 2+15, ...,8+9, the rest is symmetrical} ; Search space is only 8 pairs. From (1) generate solutions a = 7 * x^2, b =7 *y^2 see below a b 7 1792 28 1575 63 1372 112 1183 175 1008 252 847 343 700 448 567

Why m and n cant be negative numbers? For example, -1 and 18

Meanwhile my brain: √0 + √2023 = √2023

@fernandosantos3576

Жыл бұрын

sorry, 0 is not a natural number.

@mare4599

Жыл бұрын

@@fernandosantos3576 😩😩😩😩

easy. a = 2023 and b = 0

@titancac

Жыл бұрын

LETS FUCKING GOOOOOOOOO

@fernandosantos3576

Жыл бұрын

Sorry, 0 is not a natural number.

@rainyisgood1681

Жыл бұрын

@@fernandosantos3576 it is but its not positive

@fernandosantos3576

Жыл бұрын

@@rainyisgood1681, 0 is neither positive nor negative. Either way, (a,b) = (0, 2023) and (a,b)=(2023, 0) are not the only two solutions for this equation.

@rainyisgood1681

Жыл бұрын

@@fernandosantos3576 0 is a natural number i know that its not a positive or a negative number

Mindyourdecisions already solved this.

😢HUH?🤮

Dear, If I have a problem like this, using a calculator I find almost infinite possibilities of solution, so, 1) I don't see any sense in the problem; 2) I simply see a set of A and B and solve the problem: For example: a= 625 b = 399.111 25 +19.9777 = 49.9777 I do the test: a + b + 2 * SR a*b = 2023 1024,111 + 998,8888 = 2023 For me, simple as that. Happy 2023 ( look!!! 2023 !!!!!!)

@fernandosantos3576

Жыл бұрын

Only all the integers greater than zero are natural numbers.

@jezzag9739

Жыл бұрын

The solution is to solve for whole numbers only not any number

Ne saçmaydı

@mega_mango

Жыл бұрын

:(

Useless.

@ilyashick3178

Жыл бұрын

exactly

Спасибо