I love your ending message: "those who stop learning stop living". I was always into math through High School, but in college I went into another field but wound up dropping out, and I wound up being depressed with where I ended up in life, but I always came back to math by RUclips, such as your channel and others', and I really feel like keeping learning has improved my mental wellbeing.
i don't rly understand why (x-y)/y shouldn't be a fraction, for example you can raise 27 to the power 4/3 and still get an integer (81 in this case) edit: okay i've made the fix 1) you can show that x and y can be expressed as m^p and m^q, where m, p and q are integers and p > q 2) plug it into the eq and get m^(py) = m^(q(x-y) (p+q)*y = q*x (p+q) * m^q = q * m^p m^(p-q) = (p+q) / q = 1 + p/q 3) from here we get that 1 + p/q is an integer(cuz m^(p-q) is) and therefore p is a multiple of q, and that means x is a power of y
I had the same idea with 16 = 8 ^(4/3) All we can say is that k > 1 the middle section is correct and doesn't depend on k being an integer, so the conclusion n
More than one people have questioned why (x-y)/y cannot be a fraction if y^(x-y)/y is an integer. It indeed can, e.g., 4^(3/2)=8. So it needs to be proved, as follows. Write x/y in the form of a fraction: x/y=p/q, p and q are co-prime numbers. x=y^(x-y)/y=y^(p/q-1)=y^(p-q)/q. Since p-q and q are co-prime numbers because p and q are, y must be an integer to the power of q because otherwise its exponent will never cancel out the denominator q and therefore y^(p-q)/q becomes an non-integer. Let y=m^q, m€N, then x=m^q(p-q)/q=m^(p-q), x/y=m^(p-2q), therefore x/y is an integer, and (x-y)/y=x/y-1 is too an integer.
BEFORE WATCHING: Because x and y are positive integers, x^y is a positive integer. y^(x-y) must also be a positive integer, and there are two cases to consider here: Case 1: y = 1. If y = 1, then y^(x - y) = 1. x^1 = 1, so x = 1. SOLUTION FOUND: (1, 1) Case 2: x >= y. This is where it gets complicated. If x = y then y^(x - y) = 1 and by extension x^y = 1, and so actually x = 1 or y = 0. y is positive so it can't be 0, so the only other case: x = 1 then means 1 = y^(1 - y), and y = 1. already found that solution, moving on. all of the above means x > y > 1. consider the form of our equation: x^n = y^m. because all of the terms are integers, then log_y(x^n) = m, and thus log_y(x) = m/n which is rational. let b / a = m / n, where b / a is in simplest form. we can rewrite the above as log_k(x) / log_k(y) = b / a and set k^b = x and k^a = y for some positive k > 1. If b/a is in simplest form then k will turn out to be an integer. PF: given that b/a is in simplest form, suppose k isn't an integer but k^b = x. Then k is some radical of x. Let c be the smallest integer where k^c is also an integer. Then c divides b. c cannot be 1 because k isn't an integer. the same logic applies with k^a = y. c then also divides y, but then b/a wouldn't be in simplest form because they share a common factor. so by contradiction, if b/a is in simplest form then k must be an integer. back to the main quest: k^b = x, k^a = y because x > y, b > a. because y > 1, a > 0. Now the equation is (k^b)^(k^a) = (k^a)^(k^b - k^a) k^(bk^a) = k^(ak^b - ak^a) bk^a = ak^b - ak^a (b+a)k^a = ak^b 1 + b/a = k^(b - a) b > a, and k is an integer, so k^(b - a) is an integer. 1 + b/a must also be an integer, and therefore b/a must be an integer. call that integer n. b > a means n > 1. then log_y(x) = b/a => x = y^(b/a) => x = y^n Now consider the main equation yet again: (y^n)^y = y^(y^n - y) y^yn = y^(y^n - y) yn = y^n - y (safe because y > 1) n = y^(n - 1) - 1 (safe because y > 0) (n + 1) = y ^ (n - 1) (n + 1)^(1/(n - 1)) = y (safe because n > 1. Now we can start plugging in values of n. n = 2: y = 3, x = 9 9^3 = 3^(9 - 3) = 729 n = 3: y = 2, x = 8 8^2 = 2^(8 - 2) = 64 for n = 4 and above... watch this. for n = 4, n + 1 < 2^(n - 1), 5 < 8. for all n >= 4, if 2^(n - 1) > n + 1, then... 2^n = 2 * 2^(n - 1) > 2 * (n + 1) = 2n + 2 > n + 2 (n + 1) < 2^(n - 1) for all n >= 4 by induction. But also n + 1 > 1. So, 1 < n + 1 < 2^(n - 1) 1 < (n + 1)^(1/(n - 1)) < 2 There are no integers between 1 and 2, so if n >= 4 then there are no solutions. The only solutions (x, y) are (1, 1), (8, 2), and (9, 3)
Thanks for the video! I think the fastest way to arrive at the solution is recognizing that since the RHS is a power of 'y' (and must be an integer since the LHS clearly is), also the LHS must be a power of 'y'. This means that 'x' itself must be a power of 'y'. If you substutite 'x=y^n' into the original equation it simplifies to 'y^(n-1)=n+1'. This limits the possibilities a lot since the left-hand side grows generally much faster than the right-hand side. You can easily find upper bounds for 'y' and 'n' after which there cannot be any solutions to this equation. What remains is manual checking of the remaining few options for 'y' and 'n'.
@user-pl7tr9dv6l : In a case like x^2 = 9^1, the RHS is really a power of 9. This DOES NOT imply that 'x' itself must be a power of 'y'. Your assertion is true ONLY when 'y' is a prime number (or a product of prime numbers each raised to the power of 1).
Thank you for your wonderful and blessed videos, which I enjoy very much. For this problem, the reason why k = x/y must be an integer, which is important to the solution, was not obvious to me. But I think this is so because if not, k = n/m for some positive integers n and m with hcf(n, m)=1. Then the equation k = y^(k-2) becomes n/m = y^(n/m-2) hence (n/m)^m = y^(n-2m) where n-2m > 0 as k>2 RHS is an integer raised to a positive integer power and hence an integer. LHS is a non-reducing fraction raised to a positive integer power and hence still a non-reducing fraction and so not an integer. This is a contradiction. Hence k must be an integer. Again, thank you for your lovely videos and i look forward to seeing the next one. God bless you ❤
@@ciapennap900elarusenindel_9 I agree that 4^(3/2-1) is an integer but that does not disprove the fact that k>2, as you suggest. Taking x=3 and y=2 to form the power is inconsistent with then applying that power to 4. This is because the original equation (by yth rooting) is equivalent to x=y^(x/y-1). Hence the number having its power taken is y=2 not 4, so your example number is inconsistent with the problem set. I did not set out why k>2 in my previous comment. This is so for this reason. As x and y are both positive integers, so is x^y. As this equals y^(x-y) from the original equation, it follows that the power of y on RHS = x-y>0. Hence k=x/y > 1. Hence in my previous comment, n and m are both positive integers with n>m. The equation (n/m)^m = y^(n-2m) then means LHS > 1^m = 1 and hence the power of y on RHS = n-2m > 0. Hence k=n/m >2.
@@Jeremy-i1d Ok, there is more than 1 way to run into the contradiction. As you obviously deleted the part of your comment that I was referring to, I am deleting mine as well, as it does not make sense any longer. Moreover I have seen now that the fraction thing has already been dealt with long ago.
Challenging problem with a nice solution but I wonder if (x-y)/y could be a fraction since an integer raised to a rational power >1 would increase the number meaning that the inequality x>y would hold true
Loved all the constraints !!!! And love you !!! Being on a rough patch seeing your videos soothes the mind remind of my classes and the university I was free so I like to think of that time !!!!!
Wolfram Alpha gives: {{x == -8, y == -2}, {x == 1, y == -1}, {x == 1, y == 1}, {x == 8, y == 2}, {x == 9, y == 3}} which reduces to {{x == 1, y == 1}, {x == 8, y == 2}, {x == 9, y == 3}} when x and y need to be positive
there is an error at 5:50. x being bigger than y is no explanation for k being integer, 8 = 4^1.5 is a counterexample. So it needs a bit more at this part.
6:00 - this is incorrect as stated. An integer y to a fraction power can be greater than y. For example, 4^(3/2)=8 > 4. You need to use some additional/another argument here.
We have X and Y are positive integers, then x^y=y^(x-y) means x=ky where k is a positive integer. (ky)^y=y^(ky-y) (ky)^y=y^(y(k-1)) ky=y^(k-1) ky^(1-(k-1))=1 k=1 & y=1 (bc k=1 not 2) or y^(2-k)=1/k Then x=y=1✓ or y=k & 2-k=-1 k=3 and y=k=3 (x=ky) so x=9 & y=3✓ S={(1,1);(9,3)}
Before watching video x and y both are +ve integer so x^y is integer, which implies x>=Y, Else right side is non integer. 1) x = y then right side is 1, that makes only one case for left side (1^1) X AND Y both are 1 (x, y) = (1,1) 2) x > y Because both x and y is integer, and left side is x^y then right side should also be x^y but it is y^(x-y) That means y^k = x We can rewrite eqn as y^(ky) = y^(x-y) ky = x-y (k+1)y = x = y^k k = 1 - - > 2y = y (not possible) k = 2 - - > 3y = y^2 - - > y=3 & x = 9 (straightforward) k = 3 - - > 4y = y^3 - - > y = 2, x = 8 (solvable / straightforward) k = 4 If we take y = 2 then also 5y is less than 2^4 = 16, for k = 5 also 6y = 12 is less than 2^5 = 32 For all y > 2 and k >= 4 also same inequality that RHS (y^k) is greater than LHS ((k+1)y) holds and difference between RHS and LHS Will be bigger and bigger as value of k and y increases. When we take y = 1 then LHS is always higher than RHS for any value lf k So we can conclude that for k >= 4 all solution of (k+1)y = x = y^k is between 1 and 2 which are fractions or irrational numbers hence no more integer solutions. Therefore 3 possible solutions are (x, y) = (1,1), (8,2) and (9,3)
@@markcarpenter5561 if you let x = 1, then the left hand side is 1^y which is 1 for all values of y. The right hand side must equal the left hand side, so y^(1-y) = 1 for all values of y. This can only be true if y is also equal to 1. If you let y = 1, the right hand side is always equal to 1. So the left hand side x^1 is equal to 1. So x must be 1.
1:20 that is not true. You can get a negative exponent on the RHS to get a fraction and also get a fraction on the LHS if y is negative. An integer raised to an integer can definitely give you a fraction if the exponent is negative.
Not an easy problem for a 15 year old - I'll try to solve it like my 15 year old self would: You can say that it is homogeneous (dimensionally consistent) because dimensional analysis won't lead to a contradiction - e.g. assume metres for x and y. So x = ky and, because x and y are +ve integers, k must be a positive integer. (ky)^y = y^((k-1)y) = (y^(k-1))^y => ky = y^(k-1) if k = 1, x = y = 1 if k = 2, 2y = y, no solution if k = 3, 3y = y^2 => y = 3, x = 9 if k = 4, 4y = y^3 => y = 2, x = 8 if k = 5, 5y = y^4, no solution ... if k = 8, 8y = y^7, no solution and no solution for any k > 4
question: at 3:13, shouldn't x=y=0 also be a solution as then by inserting values we would get 0^0 = 0^0-0 => 0^0 = 0^0 and since LHS = RHS we can say x=y=0 is also a solution i am not good at math at all and u are the teacher so i could be wrong, also haven't watched the whole video yet. edit: ok just realized the question asks for positive values for x and y, but if it didn't, would x=y=0 be a solution?
This is a fun video, but the timing of that horn at 5:42 as you moved across the board was phenomenal. (Editing in - I may have realized that you possibly added that in yourself but it made me laugh so I'm leaving this comment lol)
If y=1, then y raised to any power is 1. Thus x^y must equal 1, and either x=1 or y=0. y is not equal to zero, so x must be 1 and therefore not greater than y.
@@nidoking042 (1,1) was the first solution. He wrote "if X >y then y >=2" when he was considering the case X>y(since X cannot be less than y). But If X>y and X cannot be 1, this means that X is >=2
@@LovePullups If you think you're disagreeing with me, you're not. You're literally saying exactly the same thing. x > y implies that both numbers are greater than one, because if either number is one then both are one. I agree that the statement should have been made in the video, but it's true nevertheless.
There’s a quick and easy proof: Take mod x of both sides: 0 = y^(x - y) mod x => y = 0 mod x Take mod y of both sides: x^y = 0 mod y => x = 0 mod y Therefore, x = y. Plugging that back in: x^x = x^(x-x) => x^x = x^0 = 1 The only positive integer solution for x^x = 1 is x = 1. Therefore (x=1, y=1) is the only solution.
@@PrimeNewtons I think the solution is basically correct and I enjoyed watching, but there is something missing with the logic as @Viki13 said in the comments " but I wonder if (x-y)/y could be a fraction since an integer raised to a rational power >1 would increase the number meaning that the inequality x>y would hold true" and someone gave an example where the exponent is not an integer: 8 = 4^(3/2).
Maths helps you to reason, use all options to get to the solution. Exactly what we do in life, you brainstorm before you undertake any project. This is where maths comes in.
I love your ending message: "those who stop learning stop living". I was always into math through High School, but in college I went into another field but wound up dropping out, and I wound up being depressed with where I ended up in life, but I always came back to math by RUclips, such as your channel and others', and I really feel like keeping learning has improved my mental wellbeing.
This testimony is a blessing. Thank you
Never Stop Teaching!
😄those who stop teaching stop living!
Bye, bye...
at 6:20 why does (x-y)/y=(x/y)-1 have to be an integer? why couldnt x/y=5/2 so (x/y)-1=3/2 means x=y^(3/2) and therefore x>y?
i don't rly understand why (x-y)/y shouldn't be a fraction, for example you can raise 27 to the power 4/3 and still get an integer (81 in this case)
edit: okay i've made the fix
1) you can show that x and y can be expressed as m^p and m^q, where m, p and q are integers and p > q
2) plug it into the eq and get m^(py) = m^(q(x-y)
(p+q)*y = q*x
(p+q) * m^q = q * m^p
m^(p-q) = (p+q) / q = 1 + p/q
3) from here we get that 1 + p/q is an integer(cuz m^(p-q) is) and therefore p is a multiple of q, and that means x is a power of y
I had the same idea with 16 = 8 ^(4/3)
All we can say is that k > 1
the middle section is correct and doesn't depend on k being an integer, so the conclusion n
x and y are integers.
4/3 is am improper fraction
had the same idea, ended up showing it for good because i didn't feel comfortable with just assuming m was gonna be an integer
More than one people have questioned why (x-y)/y cannot be a fraction if y^(x-y)/y is an integer. It indeed can, e.g., 4^(3/2)=8. So it needs to be proved, as follows. Write x/y in the form of a fraction: x/y=p/q, p and q are co-prime numbers. x=y^(x-y)/y=y^(p/q-1)=y^(p-q)/q. Since p-q and q are co-prime numbers because p and q are, y must be an integer to the power of q because otherwise its exponent will never cancel out the denominator q and therefore y^(p-q)/q becomes an non-integer. Let y=m^q, m€N, then x=m^q(p-q)/q=m^(p-q), x/y=m^(p-2q), therefore x/y is an integer, and (x-y)/y=x/y-1 is too an integer.
BEFORE WATCHING:
Because x and y are positive integers, x^y is a positive integer. y^(x-y) must also be a positive integer, and there are two cases to consider here:
Case 1: y = 1.
If y = 1, then y^(x - y) = 1. x^1 = 1, so x = 1.
SOLUTION FOUND: (1, 1)
Case 2: x >= y. This is where it gets complicated.
If x = y then y^(x - y) = 1 and by extension x^y = 1, and so actually x = 1 or y = 0. y is positive so it can't be 0, so the only other case: x = 1 then means 1 = y^(1 - y), and y = 1.
already found that solution, moving on.
all of the above means x > y > 1.
consider the form of our equation: x^n = y^m. because all of the terms are integers, then log_y(x^n) = m, and thus log_y(x) = m/n which is rational.
let b / a = m / n, where b / a is in simplest form.
we can rewrite the above as log_k(x) / log_k(y) = b / a and set k^b = x and k^a = y for some positive k > 1. If b/a is in simplest form then k will turn out to be an integer.
PF: given that b/a is in simplest form, suppose k isn't an integer but k^b = x. Then k is some radical of x. Let c be the smallest integer where k^c is also an integer. Then c divides b. c cannot be 1 because k isn't an integer.
the same logic applies with k^a = y. c then also divides y, but then b/a wouldn't be in simplest form because they share a common factor.
so by contradiction, if b/a is in simplest form then k must be an integer.
back to the main quest:
k^b = x, k^a = y
because x > y, b > a.
because y > 1, a > 0.
Now the equation is (k^b)^(k^a) = (k^a)^(k^b - k^a)
k^(bk^a) = k^(ak^b - ak^a)
bk^a = ak^b - ak^a
(b+a)k^a = ak^b
1 + b/a = k^(b - a)
b > a, and k is an integer, so k^(b - a) is an integer.
1 + b/a must also be an integer, and therefore b/a must be an integer.
call that integer n.
b > a means n > 1.
then log_y(x) = b/a => x = y^(b/a) => x = y^n
Now consider the main equation yet again:
(y^n)^y = y^(y^n - y)
y^yn = y^(y^n - y)
yn = y^n - y (safe because y > 1)
n = y^(n - 1) - 1 (safe because y > 0)
(n + 1) = y ^ (n - 1)
(n + 1)^(1/(n - 1)) = y (safe because n > 1.
Now we can start plugging in values of n.
n = 2: y = 3, x = 9
9^3 = 3^(9 - 3) = 729
n = 3: y = 2, x = 8
8^2 = 2^(8 - 2) = 64
for n = 4 and above...
watch this.
for n = 4, n + 1 < 2^(n - 1), 5 < 8.
for all n >= 4, if 2^(n - 1) > n + 1, then...
2^n = 2 * 2^(n - 1)
> 2 * (n + 1)
= 2n + 2
> n + 2
(n + 1) < 2^(n - 1) for all n >= 4 by induction.
But also n + 1 > 1.
So, 1 < n + 1 < 2^(n - 1)
1 < (n + 1)^(1/(n - 1)) < 2
There are no integers between 1 and 2, so if n >= 4 then there are no solutions.
The only solutions (x, y) are (1, 1), (8, 2), and (9, 3)
too complicated compare to his solution however its still a way to solve this
I loved this proof. I have never found such a succinct and easy-to-understand explanation, thank you! Liked and Subbed. 😀
Welcome aboard!
If a 15 y.o. kid is able to solve this, is very well trained. Congrats. nice video.
Thanks for the video! I think the fastest way to arrive at the solution is recognizing that since the RHS is a power of 'y' (and must be an integer since the LHS clearly is), also the LHS must be a power of 'y'. This means that 'x' itself must be a power of 'y'. If you substutite 'x=y^n' into the original equation it simplifies to 'y^(n-1)=n+1'. This limits the possibilities a lot since the left-hand side grows generally much faster than the right-hand side. You can easily find upper bounds for 'y' and 'n' after which there cannot be any solutions to this equation. What remains is manual checking of the remaining few options for 'y' and 'n'.
@user-pl7tr9dv6l : In a case like x^2 = 9^1, the RHS is really a power of 9. This DOES NOT imply that 'x' itself must be a power of 'y'. Your assertion is true ONLY when 'y' is a prime number (or a product of prime numbers each raised to the power of 1).
Thank you for your wonderful and blessed videos, which I enjoy very much.
For this problem, the reason why k = x/y must be an integer, which is important to the solution, was not obvious to me.
But I think this is so because if not, k = n/m for some positive integers n and m with hcf(n, m)=1. Then the equation k = y^(k-2) becomes
n/m = y^(n/m-2)
hence (n/m)^m = y^(n-2m)
where n-2m > 0 as k>2
RHS is an integer raised to a positive integer power and hence an integer.
LHS is a non-reducing fraction raised to a positive integer power and hence still a non-reducing fraction and so not an integer. This is a contradiction. Hence k must be an integer.
Again, thank you for your lovely videos and i look forward to seeing the next one. God bless you ❤
Thank you for this. I was also wondering how to conclude that (x-y)/y was a positive integer.
@@ciapennap900elarusenindel_9
I agree that 4^(3/2-1) is an integer but that does not disprove the fact that k>2, as you suggest. Taking x=3 and y=2 to form the power is inconsistent with then applying that power to 4. This is because the original equation (by yth rooting) is equivalent to x=y^(x/y-1). Hence the number having its power taken is y=2 not 4, so your example number is inconsistent with the problem set.
I did not set out why k>2 in my previous comment. This is so for this reason. As x and y are both positive integers, so is x^y. As this equals y^(x-y) from the original equation, it follows that the power of y on RHS = x-y>0.
Hence k=x/y > 1.
Hence in my previous comment, n and m are both positive integers with n>m.
The equation (n/m)^m = y^(n-2m) then means LHS > 1^m = 1 and hence the power of y on RHS = n-2m > 0. Hence k=n/m >2.
@@Jeremy-i1d Ok, there is more than 1 way to run into the contradiction. As you obviously deleted the part of your comment that I was referring to, I am deleting mine as well, as it does not make sense any longer. Moreover I have seen now that the fraction thing has already been dealt with long ago.
Challenging problem with a nice solution but I wonder if (x-y)/y could be a fraction since an integer raised to a rational power >1 would increase the number meaning that the inequality x>y would hold true
8 = 4^(3/2)
The assumption is wrong
@@aamirashraf142 the assumption still gives him answers, but maybe there would be more answers had we considered this case
@@aamirashraf142 Yes, @PrimeNewtons only showed that (x-y)/y can't be less than 1.
Sir I enjoyed your class very much, I have had a liking to mathematics but your classes are too hooking and entertaining, it was awesome
It can be reduced to:
Y = X / (log.Y(X) + 1).
Log.Y(X) must be an integer.
Therefore kY = X and Y^n = X
Y = (n+1)^(1/(n-1)) which gives solutions for n= 0,2,3
Case 2: x>=y
Let x=y+s (s is integer >0)
…
(xy)^y=y^x
…substitute and divide by y^y…
(y+s)^y=y^s
try y+s=y^(2,3,4,5,…)
etc
Belísima Questão Exponencial a duas variáveis, sua didática é Top, gostei. Parabéns Prime Newton
CLAUDIR MATTANA, LPMat. Prof.
Loved all the constraints !!!! And love you !!! Being on a rough patch seeing your videos soothes the mind remind of my classes and the university I was free so I like to think of that time !!!!!
You got this!
(xy)ʸ = yˣ
(xy)ˣʸ = y^(x²)
*TRIVIAL CASE*
xy = y => y(x - 1) = 0
xy = x² => x(x - y) = 0
y = x²
y = 0 => x = 0 [ not valid ]
*x = 1 => y = 1*
I am 14 and i love your teaching its really fun to tease my freind with this maths that you teach us
y ln (x) = (x-y) ln (y)
y ln (x) = x ln (y) - y ln (y)
y (ln (xy)) = x ln (y)
awesome video!
Wolfram Alpha gives:
{{x == -8, y == -2}, {x == 1, y == -1}, {x == 1, y == 1}, {x == 8, y == 2}, {x == 9, y == 3}}
which reduces to
{{x == 1, y == 1}, {x == 8, y == 2}, {x == 9, y == 3}}
when x and y need to be positive
Is this Wolfram premium?
@@rayyan6592 no, Wolfram Alpha
there is an error at 5:50. x being bigger than y is no explanation for k being integer, 8 = 4^1.5 is a counterexample. So it needs a bit more at this part.
Sir that's where log comes in power
log x base y +1 = x/y
6:00 - this is incorrect as stated. An integer y to a fraction power can be greater than y. For example, 4^(3/2)=8 > 4. You need to use some additional/another argument here.
Keep up the good work
Would it be possible to solve e^x=sin(x) or e^x=cos(x)? I hope the channel can solve this monstrous cases!
you are so clever!
We have X and Y are positive integers, then x^y=y^(x-y) means x=ky where k is a positive integer.
(ky)^y=y^(ky-y)
(ky)^y=y^(y(k-1))
ky=y^(k-1)
ky^(1-(k-1))=1
k=1 & y=1 (bc k=1 not 2)
or y^(2-k)=1/k
Then x=y=1✓ or y=k & 2-k=-1
k=3 and y=k=3 (x=ky)
so x=9 & y=3✓
S={(1,1);(9,3)}
@IslamDarbal After y^(2-k)=1/k , there is another possibility : y=k^(1/2) & 2-k=-2 (which leads to (k=2, y=2, x=8) or y=k^(1/3) & 2-k=-3 etc.
why does y to the power k needs to be an integer. what if k is 5/2, x>y would still be satisfied
Before watching video
x and y both are +ve integer so x^y is integer, which implies x>=Y, Else right side is non integer.
1) x = y then right side is 1, that makes only one case for left side (1^1)
X AND Y both are 1 (x, y) = (1,1)
2) x > y
Because both x and y is integer, and left side is x^y then right side should also be x^y but it is y^(x-y)
That means y^k = x
We can rewrite eqn as y^(ky) = y^(x-y)
ky = x-y
(k+1)y = x = y^k
k = 1 - - > 2y = y (not possible)
k = 2 - - > 3y = y^2 - - > y=3 & x = 9 (straightforward)
k = 3 - - > 4y = y^3 - - > y = 2, x = 8 (solvable / straightforward)
k = 4
If we take y = 2 then also 5y is less than 2^4 = 16, for k = 5 also 6y = 12 is less than 2^5 = 32
For all y > 2 and k >= 4 also same inequality that RHS (y^k) is greater than LHS ((k+1)y) holds and difference between RHS and LHS Will be bigger and bigger as value of k and y increases.
When we take y = 1 then LHS is always higher than RHS for any value lf k
So we can conclude that for k >= 4 all solution of (k+1)y = x = y^k is between 1 and 2 which are fractions or irrational numbers hence no more integer solutions.
Therefore 3 possible solutions are (x, y) = (1,1), (8,2) and (9,3)
Should not have assumed that y couldn't be 1 with x greater than 1. Other than that, excellent!
I saw that if y is 1, there's only one result for x. x must be 1.
x = 1 implies y = 1, and y = 1 implies x = 1, so x = 1 iff y = 1, and thus neither x nor y can be 1 if the other isn't also 1.
@@benkahtan6802don’t understand that. X=y=1 is a solution in the limit where x=y. Why can there not also be solutions where y=1 and x>y?
@@markcarpenter5561 if you let x = 1, then the left hand side is 1^y which is 1 for all values of y. The right hand side must equal the left hand side, so y^(1-y) = 1 for all values of y. This can only be true if y is also equal to 1.
If you let y = 1, the right hand side is always equal to 1. So the left hand side x^1 is equal to 1. So x must be 1.
@@benkahtan6802 great explanation - thank you
Oh, wow!
x = t^((t-1)/(t-2))
y = t^(1/(t-2))
t is a natural number (except 2)
1:20 that is not true. You can get a negative exponent on the RHS to get a fraction and also get a fraction on the LHS if y is negative. An integer raised to an integer can definitely give you a fraction if the exponent is negative.
Can you make a video about The Ramsey theory
i believe at "k+1 = x/y" you can use the fact that x = y*k to solve it more easily but i haven't actually tried so idk
minute 13:07 OR not AND for the two possible solutions 3 or 4.
Not an easy problem for a 15 year old - I'll try to solve it like my 15 year old self would:
You can say that it is homogeneous (dimensionally consistent) because dimensional analysis won't lead to a contradiction - e.g. assume metres for x and y.
So x = ky and, because x and y are +ve integers, k must be a positive integer.
(ky)^y = y^((k-1)y) = (y^(k-1))^y
=> ky = y^(k-1)
if k = 1, x = y = 1
if k = 2, 2y = y, no solution
if k = 3, 3y = y^2 => y = 3, x = 9
if k = 4, 4y = y^3 => y = 2, x = 8
if k = 5, 5y = y^4, no solution
...
if k = 8, 8y = y^7, no solution and no solution for any k > 4
You cant say that x/y=3 and x/y=4 are solutions before plugin them in the initial equation because you had proceed by succesive implications
what if all integers are called ?
I thought about 2^4=4^2 which doesn’t work. Nice problem
can we not use natural logs for this question?
I tried that. No big advantage.
why is (x-y)/y should be an integer. 4^3/2 is still positive even though the exponent is not an integer.
Nice
A very logical explanation and solution. Are you sure a 15-year-old can solve this problem? Well...maybe a 15-year-old math wizard.
Can they get partial credit for a partial solution? Need they prove their answer?
Or now, a 15 year-old with RUclips
@@JamesWanderswell said. Any Age that can read can learn on RUclips even 10 year olds or less if they're curious enough.
There is no way an average 15-yo can solve that lol, nice solution tho
Yeah. It's not for the average ones.
Im 16 and it was easy
Terrence Tao could’ve solved that at 7.
question:
at 3:13, shouldn't x=y=0 also be a solution
as then by inserting values we would get
0^0 = 0^0-0
=> 0^0 = 0^0
and since LHS = RHS we can say x=y=0 is also a solution
i am not good at math at all and u are the teacher so i could be wrong, also haven't watched the whole video yet.
edit: ok just realized the question asks for positive values for x and y, but if it didn't, would x=y=0 be a solution?
0 is not a positive number. x and y are stated to be positive numbers, so neither of then can be 0.
@@nanamacapagal8342 yes but IF x and y could be zero aswell, would 0 still be a solution
0⁰ is indeterminate, so zero (0, 0) is not a solution.
This is a fun video, but the timing of that horn at 5:42 as you moved across the board was phenomenal.
(Editing in - I may have realized that you possibly added that in yourself but it made me laugh so I'm leaving this comment lol)
I didn't get "if X>y then y greater or equal 2" maybe you meant X greater or equal 2"
If y=1, then y raised to any power is 1. Thus x^y must equal 1, and either x=1 or y=0. y is not equal to zero, so x must be 1 and therefore not greater than y.
@@nidoking042 (1,1) was the first solution. He wrote "if X >y then y >=2" when he was considering the case X>y(since X cannot be less than y). But If X>y and X cannot be 1, this means that X is >=2
@@LovePullups If you think you're disagreeing with me, you're not. You're literally saying exactly the same thing. x > y implies that both numbers are greater than one, because if either number is one then both are one. I agree that the statement should have been made in the video, but it's true nevertheless.
@@LovePullupsagree, but he infers that y>=2. That’s the bit I don’t understand…
There’s a quick and easy proof:
Take mod x of both sides:
0 = y^(x - y) mod x => y = 0 mod x
Take mod y of both sides:
x^y = 0 mod y => x = 0 mod y
Therefore, x = y.
Plugging that back in:
x^x = x^(x-x) => x^x = x^0 = 1
The only positive integer solution for x^x = 1 is x = 1.
Therefore (x=1, y=1) is the only solution.
At x=y^{(x-y)/y} => (x-y)/y is integer. This is wrong conclusion
👍
I like your videos, but this solution has some holes.
Highlighting the holes would really help me learn
@@PrimeNewtons I think the solution is basically correct and I enjoyed watching, but there is something missing with the logic as @Viki13 said in the comments " but I wonder if (x-y)/y could be a fraction since an integer raised to a rational power >1 would increase the number meaning that the inequality x>y would hold true" and someone gave an example where the exponent is not an integer: 8 = 4^(3/2).
x^y=y^(x-y)
x = 1, y = ± 1
x = -8, y = -2
x = 8, y = 2
x = 9, y = 3
And x = y = 0 ?
A possible solution
But it is not positive
Sir is like you are guessing values
Where do you see anything akin to a guess?
Maths helps you to reason, use all options to get to the solution. Exactly what we do in life, you brainstorm before you undertake any project. This is where maths comes in.
asnwer=2 .2/2 isit
Can you make a video about The Ramsey theory