Square Root Distance from Integers

Question

Given a decimal number k, find the smallest integer n such that the square root of n is within k of an integer. However, the distance should be nonzero - n cannot be a perfect square.

Given k, a decimal number or a fraction (whichever is easier for you), such that 0 < k < 1, output the smallest positive integer n such that the difference between the square root of n and the closest integer to the square root of n is less than or equal to k but nonzero.

If i is the closest integer to the square root of n, you are looking for the first n where 0 < |i - sqrt(n)| <= k.

Rules

• You cannot use a language's insufficient implementation of non-integer numbers to trivialize the problem.
• Otherwise, you can assume that k will not cause problems with, for example, floating point rounding.

Test Cases

.9         > 2
.5         > 2
.4         > 3
.3         > 3
.25        > 5
.2         > 8
.1         > 26
.05        > 101
.03        > 288
.01        > 2501
.005       > 10001
.003       > 27888
.001       > 250001
.0005      > 1000001
.0003      > 2778888
.0001      > 25000001
.0314159   > 255
.00314159  > 25599
.000314159 > 2534463

Comma separated test case inputs:

0.9, 0.5, 0.4, 0.3, 0.25, 0.2, 0.1, 0.05, 0.03, 0.01, 0.005, 0.003, 0.001, 0.0005, 0.0003, 0.0001, 0.0314159, 0.00314159, 0.000314159

This is code-golf, so shortest answer in bytes wins.

Answer 1

Wolfram Language (Mathematica), 34 bytes

Min[⌈.5/#+{-#,#}/2⌉^2+{1,-1}]&

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Explanation

The result must be of the form \$m^2 \pm 1\$ for some \$m \in \mathbb{N}\$. Solving the inequations \$\sqrt{m^2+1} - m \le k\$ and \$m - \sqrt{m^2-1} \le k\$, we get \$m \ge \frac{1-k^2}{2k}\$ and \$m \ge \frac{1+k^2}{2k}\$ respectively. So the result is \$\operatorname{min}\left({\left\lceil \frac{1-k^2}{2k} \right\rceil}^2+1, {\left\lceil \frac{1+k^2}{2k} \right\rceil}^2-1\right)\$.

Answer 2

Python, 42 bytes

lambda k:((k-1/k)//2)**2+1-2*(k<1/k%2<2-k)

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Based on alephalpha's formula, explicitly checking if we're in the \$m^2-1\$ or \$m^2+1\$ case via the condition k<1/k%2<2-k.

Python 3.8 can save a byte with an inline assignment.

Python 3.8, 41 bytes

lambda k:((a:=k-1/k)//2)**2-1+2*(a/2%1<k)

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These beat my recursive solution:

50 bytes

f=lambda k,x=1:k>.5-abs(x**.5%1-.5)>0 or-~f(k,x+1)

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Answer 3

05AB1E, 16 bytes

nD(‚>I·/înTS·<-ß

Port of @alephalpha's Mathematica answer, with inspiration from @Sok's Pyth answer, so make sure to upvote both of them!

Try it online or verify all test cases.

Explanation:

n                 # Take the square of the (implicit) input
                  #  i.e. 0.05 → 0.0025
 D(‚              # Pair it with its negative
                  #  i.e. 0.0025 → [0.0025,-0.0025]
    >             # Increment both by 1
                  #  i.e. [0.0025,-0.0025] → [1.0025,0.9975]
     I·           # Push the input doubled
                  #  i.e. 0.05 → 0.1
       /          # Divide both numbers with this doubled input
                  #  i.e. [1.0025,0.9975] / 0.1 → [10.025,9.975]
        î         # Round both up
                  #  i.e. [10.025,9.975] → [11.0,10.0]
         n        # Take the square of those
                  #  i.e. [11.0,10.0] → [121.0,100.0]
          TS      # Push [1,0]
            ·     # Double both to [2,0]
             <    # Decrease both by 1 to [1,-1]
              -   # Decrease the earlier numbers by this
                  #  i.e. [121.0,100.0] - [1,-1] → [120.0,101.0]
               ß  # Pop and push the minimum of the two
                  #  i.e. [120.0,101.0] → 101.0
                  # (which is output implicitly)

Answer 4

JavaScript (ES7),  51  50 bytes

f=(k,n)=>!(d=(s=n**.5)+~(s-.5))|d*d>k*k?f(k,-~n):n

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(fails for the test cases that require too much recursion)

---

Non-recursive version,  57  56 bytes

k=>{for(n=1;!(d=(s=++n**.5)+~(s-.5))|d*d>k*k;);return n}

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Or for 55 bytes:

k=>eval(`for(n=1;!(d=(s=++n**.5)+~(s-.5))|d*d>k*k;);n`)

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(but this one is significantly slower)

Answer 5

J, 39 29 bytes

[:<./_1 1++:*:@>.@%~1+(,-)@*:

NB. This shorter version simply uses @alephalpha's formula.

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39 bytes, original, brute force

2(>:@])^:((<+.0=])(<.-.)@(-<.)@%:)^:_~]

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Handles all test cases

Answer 6

Japt, 18 16 bytes

-2 bytes from Shaggy

_=¬u1)©U>½-½aZ}a

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Answer 7

Pyth, 22 21 bytes

hSm-^.Ech*d^Q2yQ2d_B1

Try it online here, or verify all the test cases at once here.

Another port of alephalpha's excellent answer, make sure to give them an upvote!

hSm-^.Ech*d^Q2yQ2d_B1   Implicit: Q=eval(input())
                  _B1   [1,-1]
  m                     Map each element of the above, as d, using:
           ^Q2            Q^2
         *d               Multiply by d
        h                 Increment
       c      yQ          Divide by (2 * Q)
     .E                   Round up
    ^           2         Square
   -             d        Subtract d
 S                      Sort
h                       Take first element, implicit print

Edit: Saved a byte, thanks to Kevin Cruijssen

Answer 8

Perl 6, 34 33 29 bytes

-1 byte thanks to Grimy

{+(1...$_>*.sqrt*(1|-1)%1>0)}

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Answer 9

APL (Dyalog Unicode), 27 bytes^SBCS

⌊/0~⍨¯1 1+2*⍨∘⌈+⍨÷⍨1(+,-)×⍨

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Monadic train taking one argument. This is a port of alephalpha's answer.

How:

⌊/0~⍨¯1 1+2*⍨∘⌈+⍨÷⍨1(+,-)×⍨ ⍝ Monadic train

                         ×⍨ ⍝ Square of the argument
                   1(+,-)   ⍝ 1 ± that (returns 1+k^2, 1-k^2)
                 ÷⍨         ⍝ divided by
               +⍨           ⍝ twice the argument
             ∘⌈             ⍝ Ceiling
          2*⍨               ⍝ Squared
     ¯1 1+                  ⍝ -1 to the first, +1 to the second
  0~⍨                       ⍝ Removing the zeroes
⌊/                          ⍝ Return the smallest

Answer 10

C# (Visual C# Interactive Compiler), 89 85 71 bytes

k=>{double n=2,p;for(;!((p=Math.Sqrt(n)%1)>0&p<k|1-p<k);n++);return n;}

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-4 bytes thanks to Kevin Cruijssen!

Answer 11

Java (JDK), 73 70 bytes

k->{double i=1,j;for(;(j=Math.sqrt(++i)%1)==0|j>=k&1-j>=k;);return i;}

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-3 bytes thanks to @ceilingcat

Answer 12

Java 8, 85 bytes

n->{double i=1,p;for(;Math.abs(Math.round(p=Math.sqrt(i))-p)>n|p%1==0;i++);return i;}

Port of EmbodimentOfIgnorance's C# .NET answer.

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The Math.round can alternatively be this, but unfortunately it's the same byte-count:

n->{double i=1,p;for(;Math.abs((int)((p=Math.sqrt(i))+.5)-p)>n|p%1==0;i++);return i;}

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Answer 13

MathGolf, 16 bytes

²_b*α)½╠ü²1bαm,╓

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Not a huge fan of this solution. It is a port of the 05AB1E solution, which is based on the same formula most answers are using.

Explanation

²                  pop a : push(a*a)
 _                 duplicate TOS
  b                push -1
   *               pop a, b : push(a*b)
    α              wrap last two elements in array
     )             increment
      ½            halve
       ╠           pop a, b, push b/a
        ü          ceiling with implicit map
         ²         pop a : push(a*a)
          1        push 1
           b       push -1
            α      wrap last two elements in array
             m     explicit map
              ,    pop a, b, push b-a
               ╓   min of list

Answer 14

Forth (gforth), 76 bytes

: f 1 begin 1+ dup s>f fsqrt fdup fround f- fabs fdup f0> fover f< * until ;

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Explanation

Starts a counter at 1 and Increments it in a loop. Each iteration it checks if the absolute value of the counter's square root - the closest integer is less than k

Code Explanation

: f                   \ start a new word definition
  1                   \ place a counter on the stack, start it at 1
  begin               \ start and indefinite loop
    1+                \ add 1 to the counter
    dup s>f           \ convert a copy of the counter to a float
    fsqrt             \ get the square root of the counter
    fdup fround f-    \ get the difference between the square root and the next closes integer
    fabs fdup         \ get the absolute value of the result and duplicate
    f0>               \ check if the result is greater than 0 (not perfect square)
    fover f<          \ bring k to the top of the float stack and check if the sqrt is less than k
    *                 \ multiply the two results (shorter "and" in this case)
  until               \ end loop if result ("and" of both conditions) is true
;                     \ end word definition

Answer 15

Jelly, 13 bytes

I have not managed to get anything terser than the same approach as alephalpha
- go upvote his Mathematica answer!

²;N$‘÷ḤĊ²_Ø+Ṃ

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How?

²;N$‘÷ḤĊ²_Ø+Ṃ - Link: number, n (in (0,1))
²             - square n        -> n²
   $          - last two links as a monad:
  N           -   negate        -> -(n²)
 ;            -   concatenate   -> [n², -(n²)]
    ‘         - increment       -> [1+n², 1-(n²)]
      Ḥ       - double n        -> 2n
     ÷        - divide          -> [(1+n²)/n/2, (1-(n²))/n/2]
       Ċ      - ceiling         -> [⌈(1+n²)/n/2⌉, ⌈(1-(n²))/n/2⌉]
        ²     - square          -> [⌈(1+n²)/n/2⌉², ⌈(1-(n²))/n/2⌉²]
          Ø+  - literal         -> [1,-1]
         _    - subtract        -> [⌈(1+n²)/n/2⌉²-1, ⌈(1-(n²))/n/2⌉²+1]
            Ṃ - minimum         -> min(⌈(1+n²)/n/2⌉²-1, ⌈(1-(n²))/n/2⌉²+1) 

Answer 16

Japt, 14 bytes

_=¬aZ¬r¹©U¨Z}a

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_=¬aZ¬r¹©U¨Z}a     :Implicit input of integer U
_                  :Function taking an integer Z as an argument
 =                 :  Reassign to Z
  ¬                :    Square root of Z
   a               :    Absolute difference with
    Z¬             :      Square root of Z
      r            :      Round to the nearest integer
       ¹           :  End reassignment
        ©          :  Logical AND with
         U¨Z       :  U greater than or equal to Z
            }      :End function
             a     :Return the first integer that returns true when passed through that function

Answer 17

Perl 5 -p, 42 bytes

$t=sqrt++$\while($p=abs$t-int$t)>$_||!$p}{

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