Difference between like powers proof

In summary, it is impossible for ##a^n - b^n = 1## for ##n > 1## and the differences between ##a^n## and ##b^n## only increase as ##n## and ##b## increase. This proof is based on the binomial expansion and the tools for this proof have existed since ancient times.
  • #1
e2m2a
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TL;DR Summary
Is the difference between like powers never equal to 1?
This may seem like a trivial question but I don't know if there is a formal proof for this. Is the following expression never true? a^n-b^n =1, where a >b, a,b,n are positive integer numbers. Was this known since ancient times? Or is there a modern proof for this?
 
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  • #2
e2m2a said:
Summary: Is the difference between like powers never equal to 1?

This may seem like a trivial question but I don't know if there is a formal proof for this. Is the following expression never true? a^n-b^n =1, where a >b, a,b,n are positive integer numbers. Was this known since ancient times? Or is there a modern proof for this?
If ##a > b##, then there is a minimum difference between ##a^n## and ##b^n##. If we fix ##b##, then the minimim difference is when ##a = b+1##. And:
$$(b+1)^n - b^n = 1 + nb + \binom n 2 b^2 + \dots nb^{n-1} > 1$$Assuming ##n > 1##, of course.
 
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  • #4
PeroK said:
If ##a > b##, then there is a minimum difference between ##a^n## and ##b^n##. If we fix ##b##, then the minimim difference is when ##a = b+1##. And:
$$(b+1)^n - b^n = 1 + nb + \binom n 2 b^2 + \dots nb^{n-1} > 1$$Assuming ##n > 1##, of course.
The OP also wanted to know whether the proof was modern. Since PeroK's proof is based on a binomial expansion, then the OP can look at https://en.wikipedia.org/wiki/Binomial_theorem#History , which at least shows how far back the tools for this proof existed.
 
  • #5
Although I posted a proof, it's clear from looking at the first few cases that ##a^n - b^n = 1## is impossible for ##n > 1##:
$$1, 4, 9, 16, \dots$$$$1, 8, 27, 64 \dots$$$$1, 16, 81, 256 \dots$$And the differences are clearly only getting larger as ##n## and ##b## increase.
 
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  • #6
PeroK said:
If ##a > b##, then there is a minimum difference between ##a^n## and ##b^n##. If we fix ##b##, then the minimim difference is when ##a = b+1##. And:
$$(b+1)^n - b^n = 1 + nb + \binom n 2 b^2 + \dots nb^{n-1} > 1$$Assuming ##n > 1##, of course.
Ok, thanks for the reply.
 
  • #7
PeroK said:
Although I posted a proof, it's clear from looking at the first few cases that ##a^n - b^n = 1## is impossible for ##n > 1##:
$$1, 4, 9, 16, \dots$$$$1, 8, 27, 64 \dots$$$$1, 16, 81, 256 \dots$$And the differences are clearly only getting larger as ##n## and ##b## increase.
Thanks for the response.
 
  • #8
PeroK said:
Although I posted a proof, it's clear from looking at the first few cases that ##a^n - b^n = 1## is impossible for ##n > 1##:
$$1, 4, 9, 16, \dots$$$$1, 8, 27, 64 \dots$$$$1, 16, 81, 256 \dots$$And the differences are clearly only getting larger as ##n## and ##b## increase.
Thanks for answering.
 

1. What is the definition of "like powers" in a mathematical proof?

Like powers refer to terms in an equation or expression that have the same base and exponent. For example, in the equation 2x^2 + 3x^2 = 5x^2, the terms 2x^2 and 3x^2 are considered like powers because they have the same base (x) and exponent (2).

2. How do you prove the difference between like powers in a mathematical proof?

To prove the difference between like powers, you must use the properties of exponents. One method is to rewrite the terms with the same base and use the property that states a^m/a^n = a^(m-n). Then, simplify the resulting expression to show that the difference between the like powers is equal to the difference of their exponents.

3. Can you provide an example of a proof for the difference between like powers?

Sure, let's say we want to prove that the difference between 5x^3 and 2x^3 is equal to 3x^3. We can rewrite 5x^3 as 2x^3 * 2 and 2x^3 as 2x^3 * 1. Using the property a^m/a^n = a^(m-n), we get (2x^3 * 2)/(2x^3 * 1) = 2x^3/2x^3 * 2/1 = 2^(3-3) * 2 = 2^0 * 2 = 1 * 2 = 2. Therefore, 5x^3 - 2x^3 = 3x^3.

4. Why is understanding the difference between like powers important in math?

Understanding the difference between like powers is important because it allows us to simplify and solve equations and expressions more efficiently. It also helps us to identify patterns and make connections between different mathematical concepts.

5. Are there any real-life applications for the difference between like powers?

Yes, the concept of like powers is used in various fields such as physics, engineering, and finance. For example, in physics, the laws of motion and gravity involve terms with like powers. In finance, compound interest and exponential growth also involve like powers. Understanding the difference between like powers can help in solving real-world problems and making predictions.

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