How does the square end?

Question

In base-10, all perfect squares end in \$0\$, \$1\$, \$4\$, \$5\$, \$6\$, or \$9\$.

In base-16, all perfect squares end in \$0\$, \$1\$, \$4\$, or \$9\$.

Nilknarf describes why this is and how to work this out very well in this answer, but I'll also give a brief description here:

When squaring a base-10 number, \$N\$, the "ones" digit is not affected by what's in the "tens" digit, or the "hundreds" digit, and so on. Only the "ones" digit in \$N\$ affects the "ones" digit in \$N^2\$, so an easy (but maybe not golfiest) way to find all possible last digits for \$N^2\$ is to find \$n^2 \mod 10\$ for all \$0 \le n < 10\$. Each result is a possible last digit. For base-\$m\$, you could find \$n^2 \mod m\$ for all \$0 \le n < m\$.

Write a program which, when given the input \$N\$, outputs all possible last digits for a perfect square in base-\$N\$ (without duplicates). You may assume \$N\$ is greater than \$0\$, and that \$N\$ is small enough that \$N^2\$ won't overflow (If you can test all the way up to \$N^2\$, I'll give you a finite amount of brownie points, but know that the exchange rate of brownie points to real points is infinity to one).

Tests:

 Input -> Output
 1     -> 0
 2     -> 0,1
 10    -> 0,1,5,6,4,9
 16    -> 0,1,4,9
 31    -> 0,1,2,4,5,7,8,9,10,14,16,18,19,20,25,28
 120   -> 0,1,4,9,16,24,25,36,40,49,60,64,76,81,84,96,100,105

this is code-golf, so standard rules apply!

(If you find this too easy, or you want a more in-depth question on the topic, consider this question: Minimal cover of bases for quadratic residue testing of squareness).

Answer 1

Pyth, 13 bytes

VQ aY.^N2Q){Y

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Lame attempt at explaining:

VQ               for N in [0 .. input-1]
                   blank space to supress implicit print inside the loop
     .^N2Q         N ^ 2 % input
   aY              append that to Y, which is initially an empty list
          )      end for
           {Y    deduplicate and implicit print

To sort the output, insert an S on any side of the {

I think there should be a shorter way...

Answer 2

Python 2, 59 bytes

t=int(input())
print list(set([(i*i)%t for i in range(t)]))

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Answer 3

PowerShell, 35 bytes

0..($n="$args")|%{$_*$_%$n}|sort -u

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Answer 4

R, 28 bytes

unique((1:(n<-scan()))^2%%n)

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Answer 5

PHP, 53 bytes

for(;$i<$t=$argv[1];)$a[$z=$i++**2%$t]++?:print"$z
";

Loop from 0 to the input number, using the n^2 mod base formula to mark numbers that have been used. It goes to that position in an array, checking if it's been incremented and outputting it if it hasn't. It then increments it afterwards so duplicate values don't get printed.

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Answer 6

8th, 138 131 bytes

Code

[] swap dup >r ( 2 ^ r@ n:mod a:push ) 1 rot loop rdrop ' n:cmp a:sort ' n:cmp >r -1 a:@ swap ( tuck r@ w:exec ) a:filter rdrop nip

Explanation

[] - Create output array

swap dup >r - Save input for later use

( 2 ^ r@ n:mod a:push ) 1 rot loop - Compute square end

rdrop - Clean r-stack

' n:cmp a:sort - Sort output array

' n:cmp >r -1 a:@ swap ( tuck r@ w:exec ) a:filter rdrop nip - Get rid of consecutive duplicates from array

SED (Stack Effect Diagram) is: a -- a

Usage and example

: f [] swap dup >r ( 2 n:^ r@ n:mod a:push ) 1 rot loop rdrop ' n:cmp a:sort ' n:cmp >r -1 a:@ swap ( tuck r@ w:exec ) a:filter rdrop nip ;

ok> 120 f .
[0,1,4,9,16,24,25,36,40,49,60,64,76,81,84,96,100,105]

Answer 7

Java (OpenJDK 8), 86 77 bytes

n->java.util.stream.IntStream.range(1,n+1).map(a->a*a%n).distinct().toArray()

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Answer 8

Pari/GP, 25 bytes

n->Set([x^2%n|x<-[1..n]])

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Answer 9

Perl 5 -MList::Util=uniq -na, 32 bytes

say for uniq map$_**2%"@F",0..$_

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Answer 10

Prolog (SWI), 49 bytes

N+X:-setof(K,A^(between(0,N,A),K is A^2mod N),X).

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Creates a set of values from K=(A^2)%N, where 0<=A<=N.

Answer 11

Pyt, 7 bytes

Đř²⇹%Ụş

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Đ            implicit input (n); Đuplicate on stack
 ř           řangify (push [1,2,...,n])
  ²          square
   ⇹%        modulo n
     Ụş      get Ụnique elements and şort in ascending order; implicit print

Answer 12

Thunno D, \$ 6 \log_{256}(96) \approx \$ 4.94 bytes

R2^$ZU

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Thunno, \$ 7 \log_{256}(96) \approx \$ 5.76 bytes

DR2^$ZU

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Explanation

DR2^$ZU  # Implicit input
DR       # Duplicate and get range(input)
  2^     # Square each number
    $    # Mod by the input
     ZU  # Uniquify
         # Implicit output

Answer 13

JavaScript, 45 bytes

n=>(g=x=>x?g(g[x*x%n]=x-1):Object.keys(g))(n)

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40 bytes

With output as an array that could contain multiple empty items.

n=>(g=x=>x?g(x-1,a[y=x*x%n]=y):a)(a=[n])

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Answer 14

J, 10 bytes

~.@:|2^~i.

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~.@:|2^~i.
        i.  NB. range [0..n)
     2^~    NB. square
  @:        NB. then
~.          NB. uniquify