Find the minimal value |ac−b|≤b

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In summary, the expression |ac−b|≤b means to find the smallest possible value of |ac−b| that is less than or equal to b. To solve for this minimal value, one can use algebraic methods to manipulate the equation and isolate the absolute value term. It is important to consider special cases, such as when b or c is equal to 0. The significance of finding this minimal value is that it helps to identify the smallest value that satisfies a certain condition, making it useful in various mathematical applications.
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lfdahl
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Find the minimal value of the expression $\frac{a}{b}$ over all triples $(a, b, c)$ of positive
integers satisfying $|a^c − b!| ≤ b$.
 
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  • #2
lfdahl said:
Find the minimal value of the expression $\frac{a}{b}$ over all triples $(a, b, c)$ of positive
integers satisfying $|a^c − b!| ≤ b$.
I guess (by instinct) the answer should be $\dfrac {1}{2}$

Am I right ?

I am thinking a method to prove it
 
  • #3
Albert said:
I guess (by instinct) the answer should be $\dfrac {1}{2}$

Am I right ?

I am thinking a method to prove it

You are right indeed! :cool:
 
  • #4
lfdahl said:
You are right indeed! :cool:
my way of thinking:
to find $min(\dfrac {a}{b})$, a must be as small as possible, if I set a=1 then c can be any positive integer
from $|a^c − b!| ≤ b $ , we know b cannot be too big,and the only solution for b is 2 and we get the answer 0.5
It is not rigorous,but give us a very quick approach
in fact c can be zero or even negative integers
 
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  • #5
Here is the suggested solution:

The answer is $\frac{1}{2}$.Note, that any triple $(a,b,c) = (1,2,c)$ satisfy the given inequality. So, $\frac{a}{b}$ takes the value

$\frac{1}{2}$. Now, we prove, that there is no other solution for $b \ge 2a$.

Let $t = |a^c-b!|$. If $t > 0$, $1 = \left | \frac{a^c}{t}-\frac{b!}{t} \right |$. Since $t \le b$, $\frac{b!}{t}$ is

an integer, so $\frac{a^c}{t}$ is also an integer. Furthermore, $2a \le b \Rightarrow a, 2a \in \left \{ 1,2,...,b \right \}$.

At least one of $a$ and $2a$ is different from $t$, so it is not canceled out from the product in

$\frac{b!}{t}$. So $a \: | \: \frac{b!}{t}$.

Therefore, since the difference between $\frac{b!}{t}$ and $\frac{a^c}{t}$ is $1$,

$gcd\left ( a,\frac{a^c}{t} \right )=1$ implying $t = a^c$. Thus, we are now left with two cases only: $\left | a^c-b! \right | = 0$ or $\left | a^c-b! \right | = a^c$. These cases reduce to: $b! = a^c$ and $b! = 2a^c$ respectively. Either way, $2a-1 \in \left \{ 1,2,...,b \right \}$,

so $ (2a-1)\: |\: b!\: |\: 2a^c$. Now, $gcd\left (2a-1, 2a^c \right )=1, \Rightarrow 2a-1= 1$ and $a = 1$.If $a = 1$, $b!-b \le 1 \Rightarrow b \le 2$. So, $(a,b) = (1,2)$ is the only solution at $b \ge 2a$. Done.

I would happily welcome a simpler proof, if anyone has a bright idea to a different approach. (Nod)
 

FAQ: Find the minimal value |ac−b|≤b

What does the expression "Find the minimal value |ac−b|≤b" mean?

The expression |ac−b|≤b means to find the smallest possible value of |ac−b| that is less than or equal to b.

How do you solve for the minimal value in the expression |ac−b|≤b?

To solve for the minimal value in the expression |ac−b|≤b, you can use algebraic methods to manipulate the equation and isolate the absolute value term. Then, you can set up inequalities to find the range of values that ac−b can take, and determine the smallest value within that range that satisfies the inequality.

Can you provide an example of solving for the minimal value in the expression |ac−b|≤b?

For example, if we have the expression |2x−3|≤5, we can isolate the absolute value by dividing both sides by 2 to get |x−3/2|≤5/2. Then, we can set up inequalities to find the range of values for x and determine the smallest value within that range that satisfies the inequality.

What is the significance of finding the minimal value in the expression |ac−b|≤b?

Finding the minimal value in the expression |ac−b|≤b can be useful in various mathematical applications, such as optimization problems or finding the roots of a polynomial function. It helps to identify the smallest value that satisfies a certain condition, which can be helpful in making decisions or solving problems.

Are there any special cases to consider when solving for the minimal value in the expression |ac−b|≤b?

Yes, there are a few special cases to consider when solving for the minimal value in the expression |ac−b|≤b. One case is when b is equal to 0, in which the minimal value would also be 0. Another case is when c is equal to 0, in which the minimal value would depend on the value of a. Careful consideration and analysis of these cases is necessary to arrive at the correct solution.

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