MHB Why do these conditions have to be satisfied?

  • Thread starter Thread starter evinda
  • Start date Start date
  • Tags Tags
    Conditions
AI Thread Summary
The discussion focuses on the process of finding solutions to the congruence equation x^2 ≡ a modulo p^n, particularly when transitioning from a known solution modulo p to finding a solution modulo p^2. It highlights that if a solution exists modulo p, it can be used to find a corresponding solution modulo p^n for n > 1. The conversation emphasizes the importance of ensuring that the new solution x_1 satisfies both the original equation and the condition x_1 ≡ x_0 modulo p. Additionally, it notes that while the existence of a solution modulo p guarantees a solution modulo p^n, the reverse is not necessarily true, as demonstrated by counterexamples. Understanding these relationships is crucial for solving congruences in number theory.
evinda
Gold Member
MHB
Messages
3,741
Reaction score
0
Hello! (Smile)

When we have a congruence $x^2 \equiv a \pmod {p^n}, n=1,2,3, \dots$ , and we know a solution $\pmod {p^n}$, then we also know a solution $\pmod {p^l}, l<n$.

For example, we know that for $n=3$, the congruence $\displaystyle{ x^2 \equiv 2 \pmod { 7^3}}$ has the solution

$$x_0 \equiv 108 \pmod {7^3} \equiv 108 \pmod {343}$$

Obviously, $x_0' \equiv 108 \pmod {49}$ is a solution of $x^2 \equiv 2 \pmod {7^2}$.

Also, $\displaystyle{ x_0'' \equiv 3 \pmod 7}$ is a solution of $x^2 \equiv 2 \pmod 7$.We want to do the reverse.

We know a solution $x_0 \pmod p$ of $x^2 \equiv a \pmod p$, and we want to find a solution $\pmod {p^2}$.
Applying this at the example $x^2 \equiv 2 \pmod 7$, we have $x_0=a_0=3$.

We are looking for a $x_1 \in \mathbb{Z}$, such that:

$$x_1^2 \equiv 2 \pmod {7^2} \text{ such that } x_1 \equiv x_0 \pmod{7}$$I haven't understood why, when we have a solution $\pmod p$, and we are looking for a solution $\pmod {p^2}$, we are looking for a $x_1$, such that:

$$x_1^2 \equiv 2 \pmod {7^2} \text{ such that } x_1 \equiv x_0 \pmod{7}$$

Could you explain it to me? (Sweating)
 
Mathematics news on Phys.org
I presume you are quoting a part of some book you have? You should snapshot the relevant page to give the readers a better idea of what is going on there (of course, only if you have a soft copy of it). From what I can exert from there, the relevant paragraph is merely a step of the whole calculations, which you have apparently omitted.

evinda said:
I haven't understood why, when we have a solution (mod p), and we are looking for a solution (mod p^2), we are looking for a x_1, such that:

Given a solution $x = x_0$ to $x^2 = a$ modulo some prime $p$, if you are looking for solutions of $x^2 = a$ modulo $p^2$, the first step would be to "sieve out" the natural numbers to look only for solutions $a \pmod{p}$ as

$$x^2 = a \pmod{p^2} \Longrightarrow x = a \pmod{p}$$

The converse doesn't hold, however! There is a lot of examples of numbers which differ modulo 2 and 4, for example. That is why I believe there is more to it than what you have posted.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top