What is the solution space for the equation 2^x - 5^y = 3 in modular arithmetic?

In summary, the possible non-negative integer pairs satisfying the equation 2^x - 5^y = 3 are (7,3), (3,1), (2,0), and (5k-1, 2k-1) where k is any integer.
  • #1
K Sengupta
113
0
Problem

Determine all possible non negative integer pairs (x, y) satisfying this equation:

2^x – 5^y = 3


My Attempt:

If x =0, then 5^y = -2, which is a contradiction.

If x =1, then 5^y =-1, which is a contradiction.

If x = 2, then 5^y = 1, so that y = 0

If x>=3, then we observe that:

3^y = 3(Mod 8), so that y must be odd

Let us substitute y = 2s+1, where s is a positive integer. …….(*)

Again, if y – y, then 2^x = 4, giving: x = 2

If y =1, 2^x = 8, giving x = 3

For y>=3, substituting y = 2s+ 1 in terms of (*), we obtain:

2^x - 5^(2s+1) = 3
Or, 2^x = 3 (Mod 5)
or, x = 4t+3, where s is a non negative integer.

So we have:

2^(4t+3) – 5^(2s+1) = 3
Or, 8*(16^t) - 5*(25^s) = 3

For t =1, we obtain s =1, so that: (x, y) = (7, 3)

Hence, so far we have obtained (x,y) = (7, 3); (3,1) and (2, 0) as valid solutions to the problem.

**** I am unable to proceed any further, and accordingly, I am looking for a methodology giving any further valid solution(s) or any procedure conclusively proving that no further solutions can exist for the given problem.
 
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  • #2
Solve it modulo ##3##.
\begin{align*}
&\phantom{\Longrightarrow} 2x-5y =3\\
&\Longrightarrow x\equiv y \mod 3 \\
&\Longrightarrow x=y+3k \\
&\Longrightarrow y=2k-1 \\
&\Longrightarrow x= 5k-1\\
&\Longrightarrow (x,y) =\{\,(5k-1,2k-1)\,|\,k\in \mathbb{Z}\,\}
\end{align*}
 

FAQ: What is the solution space for the equation 2^x - 5^y = 3 in modular arithmetic?

What is modular arithmetic and how does it work?

Modular arithmetic is a mathematical system used to calculate values within a specific set of numbers, also known as a modulus. It is based on the idea that when dividing a number by a specific modulus, the remainder is the same as the value in the set. For example, in modular arithmetic with a modulus of 5, 7 divided by 5 would have a remainder of 2, which is the same as the value in the set.

What is a modular arithmetic example?

A common modular arithmetic example is calculating the time on a 12-hour clock. In this case, the modulus is 12, and the values in the set are the numbers 1 through 12. When adding or subtracting hours, the value will wrap around once it reaches 12, just like a clock.

What are the practical applications of modular arithmetic?

Modular arithmetic has many practical applications in computer science, cryptography, and engineering. It is used in encryption algorithms, checksums, and error correction codes. It is also used in scheduling and routing problems, as well as in clock and calendar calculations.

What are the benefits of using modular arithmetic?

Modular arithmetic offers a more efficient and intuitive way to perform calculations with large numbers. It also allows for easier identification of patterns and relationships between numbers. In computer science, modular arithmetic is used to avoid overflow errors and improve the speed of calculations.

What are some common misconceptions about modular arithmetic?

One common misconception about modular arithmetic is that it only works with integers. In reality, it can be used with any type of number, including decimals and fractions. Another misconception is that modular arithmetic is only used in theoretical mathematics, when in fact it has numerous practical applications in various fields.

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