Is this number an integer power of -2?

Question

There are clever ways of determining whether a number is a power of 2. That's no longer an interesting problem, so let's determine whether a given integer is an integer power of -2. For example:

-2 => yes: (-2)¹
-1 => no
0 => no
1 => yes: (-2)⁰
2 => no
3 => no
4 => yes: (-2)²

Rules

• You may write a program or a function and use any of the standard methods of receiving input and providing output.

• Your input is a single integer, and output must be a truthy value if the integer is an integer power of -2, and a falsy value otherwise. No other output (e.g. warning messages) is permitted.

• The usual integer overflow rules apply: your solution must be able to work for arbitrarily large integers in a hypothetical (or perhaps real) version of your language in which all integers are unbounded by default, but if your program fails in practice due to the implementation not supporting integers that large, that doesn't invalidate the solution.

• You may use any programming language, but note that these loopholes are forbidden by default.

Winning condition

This is a code-golf contest: the answer which has the fewest bytes (in your chosen encoding) is the winner.

Answer 1

Mathematica, 22 bytes

EvenQ@Log2@Max[#,-2#]&

Try it online! (Using Mathics instead, where this solution also works.)

I tried to find a solution with bitwise operators for a while, and while one definitely exists, I ended up finding something which is probably simpler:

• Max[#,-2#] multiplies the input by -2 if it's negative. Multiplying by another factor of -2 doesn't change whether the value is a power of -2 or not. But now all odd powers of -2 have been turned into even powers of -2.
• But even powers of -2 are also even powers of 2, so we can use a simple Log2@... and check if the result is an integer (to check whether it's a power of 2). This already saves two bytes over Log[4,...] (another way to look at even powers of -2).
• As an added bonus, checking whether a value is an even integer is shorter than just checking whether it's an integer: we can save three more bytes by using EvenQ instead of IntegerQ.

Answer 2

Jelly, 5 bytes

æḟ-2=

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How it works

æḟ-2=  Main link. Argument: n

æḟ-2   Round n towards 0 to the nearest power of -2.
    =  Test if the result is equal to n.

Answer 3

Jelly, 6 bytes

b-2S⁼1

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This is based on how Jelly converts an integer N to any arbitrary base B, doing so by converting N to an array, in which each integer is a digit d of (N)_B, which can have a value 0≤V_d<B. Here, we will 0-index digits from the right, so every digit adds V_dB^d to form N. V_d<B⇔V_dB^d<BB^d=B^d+1, therefore every possible N has only one unique representation, if we ignore leading 0s in (N)_B.

Here, d=input, B=-2. N=B^d=1B^d=V_dB^d⇔1=V_d⇔V_d=1, and, since we're not adding any other multiples of powers of B, every other V would be 0. Right now, the array should be a 1 concatenated with d 0s. Since Jelly 1-indexes from the left, we should check whether the array's 1st element is 1, and all other elements are 0.

Hmm... all good, right? No? What's going on? Oh yeah, I have a better idea! First, let's take the sum of all of the integers in the array, treating it as if it was an integer array and not a number in base -2. If it is 1, it means that there is only one 1, and all other integers are 0. Since there can't be leading zeroes, except in the case of 0_-2 (where the sum would be 0≠1 anyways), the 1st integer must be non-zero. The only non-zero integer in the array is the 1, so it must be the first one. Therefore, this is the only case that the sum of all of the integers in the array would be 1, because the smallest possible sum of a pair of positive integers is Σ{1,1}=2, since the smallest positive integer is 1. Every integer in a base representation is non-negative, so the only way the sum is 1 is to only have one 1, and all other integers are 0. Therefore, we can just check if the sum of all of the integers in the array is 1.

Here is what the code does:

b-2S⁼1 Main link. Arguments: d
b-2    Convert d to base -2.
   S   Take the sum.
    ⁼1 Check if the sum is equal to 1.

Answer 4

Python, 46 bytes

-2 bytes thanks to @ovs.

def g(x):
 while x%-2==0!=x:x/=-2
 return x==1

Function with usage:

g(4) # put your number between the brackets

Try it online!

Answer 5

Python 2, 35 34 32 bytes

f=lambda n:n==1or n!=n%2<f(n/-2)

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Answer 6

05AB1E, 8 bytes

Y(IÄÝm¹å

Try it online! or as a Test suite

Explanation

Y(         # push -2
  IÄÝ      # push range [0 ... abs(input)]
     m     # element-wise power
      ¹å   # check if input is in the resulting list

Answer 7

JavaScript (ES6), 37 28 24 bytes

f=x=>!x|x%2?x==1:f(x/-2)

Saved 4 bytes thanks to Arnauld.

f=x=>!x|x%2?x==1:f(x/-2)

console.log(f(-2));
console.log(f(-1));
console.log(f(0));
console.log(f(1));
console.log(f(2));
console.log(f(3));
console.log(f(4));

Answer 8

Python 2, 98 50 bytes

lambda x:x*(x&-x==abs(x))*((x<0)^x.bit_length()&1)

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Answer 9

Excel, 40 36 bytes

^{Saved 4 bytes by CallumDA}

Excel can certainly do it but correcting errors adds 11 bytes

=IFERROR(-2^IMREAL(IMLOG2(A1)),1)=A1

Input is in cell A1. Output is TRUE or FALSE

If it was allowed to return either FALSE or #NUM! error for false values, it would be only 25 bytes:

=-2^IMREAL(IMLOG2(A1))=A1

Answer 10

C (gcc), 34 29 bytes

f(n){n=n%2?n==1:f(n?n/-2:2);}

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Answer 11

MATL, 9 8 bytes

2_y|:q^m

Try it online! Or verify all test cases.

How it works

Consider input -8 as an example

2_    % Push -2
      % STACK: -2
y     % Implicit input. Duplicate from below
      % STACK: -8, -2, -8
|     % Absolute value
      % STACK: -8, -2, 8
:     % Range
      % STACK: -8, -2, [1 2 3 4 5 6 7 8]
q     % Subtract 1, element-wise
      % STACK: -8, -2, [0 1 2 3 4 5 6 7]
^     % Power, element-wise
      % STACK: -8, [1 -2 4 -8 16 -32 64 -128]
m     % Ismember. Implicit display
      % STACK: 1

Answer 12

Octave, 28 bytes

@(n)any((-2).^(0:abs(n))==n)

This defines an anonymous function. The approach is similar to that in my MATL answer.

Try it online!

Answer 13

Regex (ECMAScript / Java), 38 37 34 33 bytes

\b(?=(^|x))((x*)\3\3(?=\3$))*x\1$

Try it online! - ECMAScript
Try it online! - Java

Takes its input in unary, as an optional - sign followed by a string of x characters, the count of which represents the absolute value of the number. (As such, it is not bijective unary, as zero can be represented in two ways. The regex works with both.)

Sets a flag for whether a negative sign is present. Operating on the absolute value, repeatedly divides by 4 (requiring that there is no remainder), and then asserts that the remaining number is 1 if there was no negative sign, or 2 if there was.

\b                    # Anchor on the border between {start or "-"} and "x"; in the
                      # domain of this problem, it is actually equivalent to
                      # (?<=^(?=x)|-|x$) - so it can anchor on the end instead of the
                      # beginning, but that would result in a non-match.
(?=(^|x))             # \1 = 1 if negative sign is present, 0 if not
(                     # loop the following:
    (x*)\3\3(?=\3$)   # assert tail is divisible by 4; tail = tail / 4
)*                    # iterate as many times as possible (minimum 0)
x                     # tail -= 1
\1                    # tail -= \1
$                     # assert tail==0

Regex (ECMAScript), 36 bytes

^(-(x*)(?=\2$))?((x*)\4\4(?=\4$))*x$

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This was the original version I posted; I'm keeping it here, as it's sufficiently different, yet close enough in length, to be interesting.

If it sees a negative sign, it strips it and divides the number by 2 (requiring that there is no remainder). It then asserts that the remaining number is a power of 4.

^
(
    -                 # eat a leading negative sign
    (x*)(?=\2$)       # assert tail is even; tail = tail / 2
)?                    # do the above optionally
# Assert that tail is a power of 4
(                     # loop the following:
    (x*)\4\4(?=\4$)   # assert tail is divisible by 4; tail = tail / 4
)*                    # iterate as many times as possible (minimum 0)
x$                    # assert tail == 1; if this fails to match, the regex engine will
                      # try backtracking the loops, but that cannot result in a match

Regex (Perl / PCRE / Boost / Python / Ruby / .NET), 32 bytes

^(-)?((x*)\3\3(?=\3$))*x(?(1)x)$

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Uses essentially the same algorithm as the 33 byte ECMAScript version, but in a more straightforward way, which never matches strings outside the problem's domain.

^
(-)?                  # \1 = negative sign if it is present; unset otherwise
(                     # loop the following:
    (x*)\3\3(?=\3$)   # assert tail is divisible by 4; tail = tail / 4
)*                    # iterate as many times as possible (minimum 0)
x                     # tail = tail - 1
(?(1)x)               # if \1 is set, then tail = tail - 1, else do nothing
$                     # assert tail == 0

---

Regex (ECMAScript or better), 26 bytes

^((.*)\2\2(?=\2$))*(x|oo)$

Try it online! - ECMAScript
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Try it online! - PCRE2
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Takes its input in bijective signed unary. A nonnegative number is a string of x characters, and a nonpositive number is a string of o characters. The the length of the string is the absolute value of the number.

^                     # tail = N = input number
(                     # Loop the following:
    (.*)\2\2(?=\2$)   # Assert tail is divisible by 4; tail = tail / 4;
                      # use "." so that either "x" or "o" can be matched, as
                      # long as all characters are identical.
)*                    # Iterate as many times as possible (minimum 0).
(x|oo)$               # Assert tail == 1 or tail == -2

Answer 14

HP 28s 36 bytes

WHILE DUP ABS 1 > REPEAT 2 NEG / END 1 ==

Answer 15

PHP, 41 Bytes

for(;$argn%-2==0;)$argn/=-2;echo$argn==1;

PHP, 52 Bytes

echo($l=log(abs($argn),2))==($i=$l^0)&&$argn>0^$i%2;

PHP, 64 Bytes

Working with a Regex

echo preg_match("#^".($argn>0?1:"1+0")."(00)*$#",decbin($argn));

Answer 16

JavaScript (ES6), 21 bytes

A recursive function that returns 0 or true.

f=n=>n==1||n&&f(n/-2)

How it works

This doesn't include any explicit test -- like n being odd or abs(n) being less than one -- to stop the recursion early when the input is not an exact power of -2.

We exit only when n is exactly equal to either 1 or 0.

This does work however because any IEEE-754 float will eventually be rounded to 0 when divided by 2 (or -2) enough times, because of arithmetic underflow.

Test cases

f=n=>n==1||n&&f(n/-2)

console.log(f(-2));
console.log(f(-1));
console.log(f(0));
console.log(f(1));
console.log(f(2));
console.log(f(3));
console.log(f(4));

Answer 17

C (gcc), 33 bytes

f(n){return n%2?n==1:n&&f(n/-2);}

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Answer 18

Python, 24 bytes

lambda n:n*n&n*n-1<n%3%2

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The bit trick k&k-1==0 checks whether k is a power of 2 (or k==0). Checking this for k=n*n as n*n&n*n-1==0 tells us whether abs(n) is a power of 2.

To further see if n is a power of -2, we need only check that n%3==1. This works because mod 3, the value -2 is equal to 1, so its powers are 1. In contrast, their negations are 2 mod 3, and of course 0 gives 0 mod 3.

We combine the checks n*n&n*n-1==0 and n%3==1 into a single expression. The first can be written with <1 for ==0, since it's never negative. The n%3==1 is equivalent to n%3%2, giving 0 or 1. So, we can combine them as n*n&n*n-1<n%3%2.

Answer 19

05AB1E, 5 4 bytes

Ties emanresu A's Vyxal answer for #1.

2(вO

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2(вO  # full program
   O  # push sum of...
  в   # list of base-10 values of base...
 (    # negative...
2     # literal....
  в   # digits of...
      # implicit input
      # implicit output

Answer 20

Python 3, 34 bytes

lambda n:n==(-2)**~-n.bit_length()

Answer 21

Java 7, 55 bytes

boolean c(int n){return n==0?0>1:n%-2==0?c(n/-2):n==1;}

Explanation:

boolean c(int n){  // Method with integer parameter and boolean return-type
  return n==0 ?    //  If n is zero:
    0>1//false     //   Return false
   : n%-2==0 ?     //  Else-if n mod -2 is zero:
    c(n/-2)        //   Recursive call for the input divided by -2
   :               //  Else:
    n==1;          //   Return if n is one
}                  // End of method

Test code:

Try it here.

class M{
  static boolean c(int n){return n==0?0>1:n%-2==0?c(n/-2):n==1;}

  public static void main(String[] a){
    for(int i = -2; i <= 4; i++){
      System.out.println(i + ": " + c(i));
    }
  }
}

Output:

-2: true
-1: false
0: false
1: true
2: false
3: false
4: true

Answer 22

Perl 6, 21 bytes

{$_==(-2)**(.lsb//0)}

Try it

Expanded:

{  # bare block lambda with implicit parameter ｢$_｣

  $_                  # is the input
  ==                  # equal to
  (-2)**( .lsb // 0 ) # -2 to the power of the least significant bit of the input
}

Note that 0.lsb returns Nil which produces a warning when used as a number, so the defined or operator // is used.
(Think of // as || with a different slant)

A method call with no invocant where a term is expected is implicitly called on $_. (.lsb)

Also works with .msb.

Answer 23

R, 22 bytes

Takes input from stdin, returns TRUE or FALSE accordingly.

scan()%in%(-2)^(0:1e4)

I'm not 100% sure that this is a valid answer, as it only works for integers up to R's size limit, and if the integers were unbounded it wouldn't work. However, the rules state:

The usual integer overflow rules apply: your solution must be able to work for arbitrarily large integers in a hypothetical (or perhaps real) version of your language in which all integers are unbounded by default, but if your program fails in practice due to the implementation not supporting integers that large, that doesn't invalidate the solution.

In a hypothetical version of R which does allow unbounded integers, then we could use the following code, for the same byte count:

scan()%in%(-2)^(0:Inf)

Of course, in real R, the above code just gives Error in 0:Inf : result would be too long a vector.

Answer 24

Haskell, 24 23 bytes

f 0=0
f 1=1
f n=f(-n/2)

Defines a function f which returns 1 for powers of -2 and 0 otherwise.

A golfed version of my first submission to the other challenge.

Answer 25

Julia 0.5, 20 bytes

!n=n∈(-2).^(0:n^2)

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Answer 26

Pyth, 7 bytes

!tsjQ_2

Test suite

Explanation:

     _2   # -2
   jQ     # Cast input to that base
          # Iff input is a power of -2, then jQ_2 returns something of the form [1]+[0]*n
          # where n is the power of -2
          # This is also the only situation in which the sum of that is 1
!ts       # so we check for that: is the (s)um equal to 1 (or, equivalently, "!(sum - 1)" )

Answer 27

Risky, 14 bytes

__-+_-2{[?+_0:__0+_0+__?+_0

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Explanation:

• __-+_-: -2, written as -1 + -1 with padding
• 2: Power
• {[?+_0: log_2(abs(input)), with +_0 as padding
• :: Equals
• __0+_0+__?+_0: Input (?), with padding

So basically, -2 ** log_2(abs(input)) == input

Answer 28

x86-32 machine code, 20 18 16 bytes

-2 bytes thanks to @xiver77 -2 bytes by removing an unnecessary jump

(This submission was originally 19 bytes, but I adjusted it to 20 since I forgot to add ret).

00000000: f8a9 ffff 7f00 7507 c1f8 1730 e0d1 f8c3  ......u....0....

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Expects a single precision float in eax. Output is in the carry flag.

It takes advantage of some fundamental properties of floats, for some power of -2 n:

• abs(n) is a power of 2, therefore n's mantissa must be 0.
• n is positive for odd numbered exponents, and negative for even exponents. In other words, sign ^ exponent[0] = 1 where exponent[0] is the least significant bit of the exponent.

NASM source:

[BITS 32]

isnegpow2:
    ; Clear carry
    clc
    ; Check if mantissa is non-zero
    test eax, 0x7fffff
    ; If it is, then abs(eax) is not a
    ; power of 2, so eax cannot be a power
    ; of -2
    jnz .end

    ; Shift away the mantissa
    sar eax, 23
    ; Now ah contains the sign,
    ; and al contains the exponent.
    ; For powers of -2, sign ^ exponent[0] = 1
    xor al, ah
    ; Check lsb by shifting it to the carry flag
    ; If it's 1, then eax is a power of -2, so the
    ; carry will be set
    sar eax, 1
.end:
    ret

Answer 29

Javascript(ES7), 45 bytes

x=>-1**Math.log2(Math.abs(x))*Math.abs(x)==x

Answer 30

Prolog (SWI), 44 bytes

p(X):-X=1;X\=0,X mod 2=:=0,Z is X/(-2),p(Z).

Online interpreter