What else could we do? (p-adic expansion)

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In summary, when trying to find the p-adic expansion of fractions 1/p and 1/p^r in the field of p-adic numbers, it is correct that the congruences do not have solutions. This is because the digit list for these fractions would have non-zero digits only in positions with negative indices, which is not allowed in p-adic integers. Instead, the digit list would have all zeros except for the digit corresponding to the negative power of p, which would be 1. This can be seen in examples such as 1/5 and 1/5^3 in the 5-adic field.
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evinda
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Hello! (Wave)

I want to find the p-adic expansion of $\frac{1}{p}$ and $\frac{1}{p^r}$ in the field $\mathbb{Q}_p$.

So, do I have to solve the congruences $px \equiv 1 \pmod {p^n}, p^r x \equiv 1 \pmod { p^n }, \forall n \in \mathbb{N} $, respectively?

But.. these congruences do not have solutions, right? (Thinking)

What else could we do, in order to find the p-adic expansion of $\frac{1}{p}$ and $\frac{1}{p^r}$ in $\mathbb{Q}_p$ ? (Worried)
 
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If you are trying to express 1/p and 1/p^r as a list of p-adic integer digits,
{a0, a1, a2, ... }
then it is correct " these congruences do not have solutions"

These are not p-adic integers. The digit list has non-zero an, for n<0.

Case 1/p:
all digits zero except a-1 = 1

Case 1/p^r:
all digits zero except a-r = 1

Examples: 1/5 and 1/5^3 as 5-adic
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Related to What else could we do? (p-adic expansion)

1. What is p-adic expansion?

P-adic expansion is a mathematical concept that involves representing numbers in a different base system than the traditional base 10. Instead, it uses a prime number, p, as its base. This type of expansion is commonly used in number theory and has applications in various areas of mathematics and physics.

2. How does p-adic expansion differ from decimal expansion?

P-adic expansion differs from decimal expansion in several ways. Firstly, it uses a prime number as its base, whereas decimal expansion uses base 10. Additionally, in p-adic expansion, the digits to the left of the decimal point represent increasingly smaller fractions, while in decimal expansion, the digits to the right of the decimal point represent increasingly smaller fractions.

3. What are some real-world applications of p-adic expansion?

P-adic expansion has applications in various areas of mathematics and physics. For example, it is used in cryptography to generate secure prime numbers. It also has applications in number theory, such as in the study of prime numbers and their properties. In physics, p-adic numbers are used in string theory and p-adic analysis is used to study the behavior of particles in quantum mechanics.

4. How does p-adic expansion relate to the concept of infinity?

P-adic expansion is closely related to the concept of infinity. In traditional decimal expansion, the digits to the right of the decimal point continue infinitely, whereas in p-adic expansion, the digits to the left of the decimal point continue infinitely. This reflects the different ways in which p-adic and decimal numbers approach infinity.

5. Can p-adic expansion be used to solve certain mathematical problems more efficiently?

Yes, in some cases, p-adic expansion can be used to solve mathematical problems more efficiently. For example, in some cases, calculations involving p-adic numbers can be done faster than calculations involving decimal numbers. P-adic numbers also have properties that can make certain mathematical problems easier to solve, such as in the study of prime numbers and their properties.

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