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anemone
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Prove that $\large 3^{4^5}+4^{5^6}$ is the product of two integers, each at least $\large 10^{2009}$.
anemone said:Prove that $\large 3^{4^5}+4^{5^6}$ is the product of two integers, each at least $\large 10^{2009}$.
Prove that $3^{4^5}+4^{5^6}$ is the product of two integers,
each at least $\large 10^{2009}$.
soroban said:Hello, Albert!
You are misreading the exponents.
In an exponential "stack",
. . we read from the top down.
. . [tex]3^{4^5} \;=\;3^{1024}[/tex]
. . [tex]4^{5^6} \;=\;4^{15,625}[/tex]However: .[tex](3^4)^5 \;=\;8^4\:\text{ and }\: (4^5)^6 \:=\:1024^6[/tex]
Albert said:thanks soroban , in a haste I made a mistake in misreading the exponent
now the solution is as follows :
$(3^{512})^2+(2^{15625})^2---(1)$
let $a=3^{512}, b=2^{15625}$
(1) becomes $(a+b)^2 -2ab=(a+b)^2 -[(3^{256}\times 2^{7813})]^2=(x+y)(x-y)$
$here \,\, x=a+b, y=(3^{256}\times 2^{7813}) $
the rest is easy:
we only have to compare x-y and $10^{2009}$(compare digit numbers of both values)
I only count them roughly
$15625\times log 2>15625\times 0.3>4687>2009$
$256\times log 3+7813\times log 2>102+2343$
4687-102-2343=2242>2009
$\therefore x-y >10^{2009}$
and the proof is finished
A product of two integers is the result of multiplying two whole numbers together. For example, the product of 3 and 4 would be 12.
To find the product of two integers, simply multiply the two numbers together. If the integers are both positive, the product will also be positive. If one integer is negative, the product will be negative. If both integers are negative, the product will be positive.
No, the product of two integers will always be a whole number. If the result of multiplying two integers is a fraction or decimal, it is not considered a product of two integers.
The product of two integers is the result of multiplication, while the sum of two integers is the result of addition. For example, the product of 3 and 4 is 12, while the sum of 3 and 4 is 7.
Yes, the product of two integers can be negative if one or both of the integers is negative. For example, the product of -3 and 4 is -12.