Existence of a non-square integer L such that L is a quadratic residue modulo p n for all n.The background of this question is on the sequences in p-adic integers, in which I originally looked into the following problem:Can a sequence of perfect squares converge to a non-square value under the p-adic metric?Where the p-adic metric is, as usual, defined as d p ( m , n ) = p − max { n ∈ N : p n | ( m − n ) ) } For m ≠ n, and d p ( m , m ) := 0Obviously, it is ask for the existence of a function f : N → Z and a non-square number L such that for all n ∈ N , f ( n ) 2 − L = p g ( n ) q ( n )Where g : N → N is an unbounded function and q(n) is coprime to n for all n. Clearly, this is also equivalent to say that the equation x 2 ≡ L ( mod p n )Is soluble for infinitely many n. But we know that if L is a quadratic residue modulo p n , it is also a quadratic residue modulo p n − 1 . Therefore, the statement resolves to the following proposition:There is (not) a non-square integer L such that it is a square modulo p n for every n ∈ N .Which I know neither is true or not, nor how to start off decently. Though I have not found any of them, and I believe that such L does not exist, i still cannot find out a complete proof. The proposition can indeed be generalized into the followingThere is (not) a non-kth power L such that it is congruent to a kth power modulo p n for all n ∈ N

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Existence of a non-square integer L such that L is a quadratic residue modulo       p    n   for all n.The background of this question is on the sequences in p-adic integers, in which I originally looked into the following problem:Can a sequence of perfect squares converge to a non-square value under the p-adic metric?Where the p-adic metric is, as usual, defined as      d    p    (  m  ,  n  )  =      p          −      max      {      n      ∈              N            :              p        n                    |            (      m      −      n      )      )      }      For   m  ≠  n, and       d    p    (  m  ,  m  )  :=  0Obviously, it is ask for the existence of a function   f  :      N    →      Z   and a non-square number L such that for all   n  ∈      N  ,  f  (  n      )    2    −  L  =      p          g      (      n      )        q  (  n  )Where   g  :      N    →      N   is an unbounded function and q(n) is coprime to n for all n. Clearly, this is also equivalent to say that the equation      x    2    ≡  L    (  mod        p    n    )Is soluble for infinitely many n. But we know that if L is a quadratic residue modulo       p    n  , it is also a quadratic residue modulo       p          n      −      1      . Therefore, the statement resolves to the following proposition:There is (not) a non-square integer L such that it is a square modulo       p    n   for every   n  ∈      N  .Which I know neither is true or not, nor how to start off decently. Though I have not found any of them, and I believe that such L does not exist, i still cannot find out a complete proof. The proposition can indeed be generalized into the followingThere is (not) a non-kth power L such that it is congruent to a kth power modulo       p    n   for all   n  ∈      N

erlentzed · Accepted Answer

Step 1Not only are there plenty of non-squares which become squares in the p-adic numbers             Z        p  , the phenomena underlying this was one of the motivating reasons for constucting the p-adic numbers in the first place. Here are some hints (only hints!) to get you started.Notice that      2    2    =  −  1    mod      5  ,  (  2  +  5      )    2    =      7    2    =  −  1    mod          5    2    ,  (  2  +  5  +  2  ⋅      5    2        )    2    =      57    2    =  −  1    mod          5    3    .Step 2So now try the following:A. Modify 57 by adding a multiple of       5    3    =  125 so that   (  57  +  k  ⋅      5    3        )    2    ≡  −  1    mod          5    4  B. Inductively construct       x    n   with the following properties.1.       x    1    =  22.       x    n    ≡      x          n      −      1          mod          5          n      −      1      3.       x    n    2    ≡  −  1    mod          5          n      C. Think about how general the construction above is. What would happen if one replaces -1 and 5 by a and p, where a is a square modulo p, and p is an odd prime. What happens when   p  =  2?

Existence of a non-square integer L such that L is a quadratic residue modulo p^n for all n. Can a sequence of perfect squares converge to a non-square value under the p-adic metric?

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