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lfdahl
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Determine the positive numbers, $a$, such that the sum:
$$\sqrt[3]{3+\sqrt{a}}+\sqrt[3]{3-\sqrt{a}}$$
is an integer.
$$\sqrt[3]{3+\sqrt{a}}+\sqrt[3]{3-\sqrt{a}}$$
is an integer.
my solution:lfdahl said:Determine the positive numbers, $a$, such that the sum:
$$\sqrt[3]{3+\sqrt{a}}+\sqrt[3]{3-\sqrt{a}}$$
is an integer.
Albert said:my solution:
let $\sqrt[3]{3+\sqrt{a}}=x$
$\sqrt[3]{3-\sqrt{a}}=y$
consider $x>0,y>0$
$x^3+y^3=(x+y)(x^2-xy+y^2)=6=2\times 3=3\times 2=6\times 1=1\times 6$
only $x+y=2---(1)$
and $x^2-xy+y^2=3$ $\rightarrow xy=\dfrac {1}{3}---(2)$
satisfy the condition $a>0$
$x,y$ are roots of $3t^2-6t+1=0$
$x=\dfrac{3+\sqrt 6}{3}$
$y=\dfrac{3-\sqrt 6}{3}$
and we get :$a\approx 8.963$
if $x+y<0$
$x^3+y^3=(x+y)(x^2-xy+y^2)=6=-2\times -3=-3\times -2=-6\times -1=-1\times -6$
with the same method we may get another "a" or no solution
another solution:lfdahl said:Would you please show the other $a$ solution?
Albert said:another solution:
sorry another solution will happen at :
$x+y=1$ and
$x^2-xy+y^2=6,or \,\, 3xy=-5$
which gives $x=1.8844,or\, -0.8844$
both yield $a=13.63$
here ($\sqrt[3]{3+\sqrt a}=x$)
($\sqrt[3]{3-\sqrt a}=y$)
The formula for (3+√a)^(1/3)+(3-√a)^(1/3) is (3+√a)^1/3 + (3-√a)^1/3 = 2∛(3a+√(9a^2-4))/∛(3a+√(9a^2-4)).
(3+√a)^(1/3)+(3-√a)^(1/3) is an integer if the expression inside the square root (√(9a^2-4)) is a perfect square. This means that the value of a must be such that 9a^2-4 is a perfect square.
The significance of (3+√a)^(1/3)+(3-√a)^(1/3) being an integer is that it indicates the presence of a special type of Pythagorean triple, where the sum of two cube roots is also an integer. This has interesting implications in number theory and can lead to further discoveries.
To find the value of a that makes (3+√a)^(1/3)+(3-√a)^(1/3) an integer, we need to solve the equation 9a^2-4 = b^2, where b is an integer. This can be done by trial and error or by using algebraic methods such as completing the square or quadratic formula.
One real-world application of (3+√a)^(1/3)+(3-√a)^(1/3) being an integer is in the field of cryptography. The expression can be used to generate secret keys for encryption schemes, as the presence of a special Pythagorean triple makes the key more secure and difficult to crack.