# Is this number an integer power of -2?

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 contest: the answer which has the fewest bytes (in your chosen encoding) is the winner.

• @KritixiLithos I don't see why it should. There is no integer i such that (-2)^i = 2 Apr 6 '17 at 12:10
• Are the exponents positive or -0.5 should be valid since it's 2^(-1). Apr 6 '17 at 12:13
• @Mr.Xcoder, Since inputs are always integer values, a negative exponent won't be required (or possible). Apr 6 '17 at 12:16
• @SIGSEGV maybe whereas i is not natural Apr 6 '17 at 12:55
• @Jason, as many as supported/natural in your language - see the third rule. And it's code-golf because it needs an objective winning criterion to be on-topic here - "a pleasing solution" doesn't cut it (though I do like the Mathematica answer - that surprised me). Apr 6 '17 at 14:55

# bc 88 bytes

bc -l <<< "n=$1;q=l(sqrt(n*n));p=4*a(1);((n<1)*c(q/l(2)*p/2)+(n>1)*(s(q/l(4)*p)))^2==0"  I have this in a file neg2.sh and it prints 1 for powers of -2 and 0 otherwise I know it's really long, but it was fun ### Test $ for i in {-129..257}; do echo -n "$i: "; ./neg2.sh$i; done | grep ': 1'
-128: 1
-32: 1
-8: 1
-2: 1
1: 1
4: 1
16: 1
64: 1
256: 1


### Explanation

The main body has two halves, both are trying to equal zero for powers of -2.

q=l(sqrt(n*n))               % ln of the absolute value of the input
p=4*a(1)                     % pi: arctan(1) == pi/4
q/l(2) -> l(sqrt(n*n))/l(2)  % change of base formula -- this gives
% the power to which 2 is raised to equal
% sqrt(n*n). It will be an integer for
% numbers of interest
n<1                          % 1 if true, 0 if false. for negative
% numbers check for powers of 2
n>1                          % for positive numbers, check for powers
% of 4
c(q/l(2)*p/2)                % cos(n*pi/2) == 0 for integer n (2^n)
s(q/l(4)*p)                  % sin(n*pi) == 0 for integer n (4^n)
(....)^2==0                  % square the result because numbers are
% not exactly zero and compare to 0

• I never expected trigonometry! Good answer! Apr 11 '17 at 8:24

## Alice, 12 bytes, non-competing

/o|\ntzR2
@i


Try it online!

### Explanation

Alice has a fairly weird built-in, which was added because I needed something that goes well thematically with the string operation "discard everything up to this substring". That operation is "drop small factors" and what it does for positive x and y is that it divides x by all of its prime factors less than or equal to y. But if y is negative, then Alice tries negative prime factors greater than or equal to y instead, which means that every time a prime factor is removed, the sign of x changes. So if we use -2 as the second argument, we'll end up with 1 if and only if the input is a power of -2 (if the input is not a power of two, other factors will remain in the end, and if it has the wrong sign, we'll end up with -1 instead of 1).

The rest of the program is just a bit of weird control flow.

/   Reflect southeast. Switch to Ordinal.
i   Read all input as a string.
Reflect off boundary, move northeast.
|   Reflect northwest.
Reflect off boundary, move southwest.
i   Read all input as a string, but there's no input left, so this pushes "".
Reflect off boundary, move northwest.
/   Reflect west. Switch to Cardinal.
Wrap around to the end of line 1.
2R  Push -2. (Really: push 2, negate.)
z   Drop small factors. When trying to find a second integer argument,
this discards the empty string and then implicitly converts the input
string to an integer. Turns only valid inputs to 1.
tn  Decrement, logical NOT. Effectively an "equals 1?" check.
\   Reflect southwest. Switch to Ordinal.
Reflect off boundary, move northwest.
o   Implicitly convert result to a string and print it.
Reflect off boundary, move southwest.
@   Terminate the program.


# Fourier, 53 bytes

I~X1~N~G0(0-2*G~GX*X~PG*G>P{1}{0~O~N}G{X}{1~O0~N}N)Oo


I'll work on golfing this later, but the outline of this is:

X = User input
G = N = 1
Loop until N = 0
G = -2 * G
P = X*X
If G*G > P then
N = O = 0
End if
If G = X then
O = 1
N = 0
End if
End loop
Print O


Where the output is 0 for falsey and 1 for truthy.

Try it online!

• In the algo description not would be better not use P variable and write If G*G > X*X then...?
– user58988
Apr 12 '17 at 15:13
• @RosLuP That would be better, but Fourier would simply treat that as (G*G > X)*X Apr 12 '17 at 19:44

# Casio BASIC, 76 bytes

Note that 76 bytes is what it says on my calculator.

?→X
0→O
While Abs(X)≥1
X÷-2→X
If X=1
Then 1→O
IfEnd
WhileEnd
O


This is my first venture into Casio BASIC... I never realised I could write such decent programs on a calculator :D

# Clojure, 57 bytes

(defn i[n](if(= n 1)true(if(=(int n)0)false(i(/ n -2)))))


Try it online!

Full function, with annotations:

;; Define function is-pow? with 1 argument, n
(defn is-pow? [n]
;; If n = 1, that means it's a power of -2,
;; so we return true
(if (= n 1) true
;; When we recursively call the function,
;; -1 > n > 1. int rounds up when the number
;; is negative ((int -1/2) = 0), and rounds down
;; when the number is positive. It also catches
;; the edgecase of 0.
(if (= (int n) 0) false
;; If n made it to here, n < -1 or n > 1 - we have
;; to call the function recursively.
(is-pow? (/ n -2)))))


# 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 +*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)" )


# Perl 5-p, 32 bytes

$.*=-2while(abs)>abs$.;$_=$.==$_  Try it online! # Rust, 41 bytes |x|(|mut e|{while e*e<x*x{e*=-2}e==x})(1)  Try it online! # Common Lisp, 51 bytes (defun f(n)(or(= 1 n)(and(< 1(abs n))(f(/ n -2)))))  Recursive version. Try it online. • Try it online! Mar 22 at 6:05 • @Deadcode , added, thanks. Mar 22 at 7:06 # Python 3.8 (pre-release), 39 bytes lambda n:len(x:=bin(n))&1==x.count('1')  Try it online! # Vyxalr, 6 bytes ȧʀudec  Try it Online! Very rude, I know, but it gets the job done. ## Explained ȧʀudec ȧʀ # range(0, abs(input) + 1) ud # -2 (-1 * 2). 2N would have worked just the same, but that wouldn't be as funny now would it. e # -2 ** the range (vectorises) c # is input in ^  • r is for rude Jul 20 at 16:42 ## Python 2.7, 40 bytes a=input() while a%-2==0:a/=-2 print a==1  Credits to Mr. Xcoder for the original code of length 43 bytes. Had to post as a separate answer since I don't have enough reputation to comment. • It's kind of the same thing, since I've made my answer version-universal, so it works in both Python 2 and 3. If you were to do this in Python 3, you should have added int(input()) which would have gone over the limit of the def-like function. Additionally, In python 3, you must use print() which would of wasted 1 byte. That's why I chose that way, because in Python 3 it gets longer... Apr 6 '17 at 15:27 # Cjam, 12 bytes li_z2mLi-2#=  Explanation comes later. • I think you'll have to use 2b, instead of 2mL, because CJam does have arbitrary-precision integers, so your answer should work for arbitrarily large inputs (but mL won't be able to handle inputs correctly that can't be represented exactly as a 64-bit float). That should save bytes anyway, because you need neither the z, nor the i (although you will need to decrement the result before doing the exponentiation). Apr 9 '17 at 18:16 # Javascript ES6, 51 50 chars Not very short, but I hope interesting :) x=>eval(for(w=q=1;w<=(x<0?-x:x);w*=2,q*=-2)q==x)  Test: f=x=>eval(for(w=q=1;w<=(x<0?-x:x);w*=2,q*=-2)q==x) for(x=-2049; x<2049; ++x) if(f(x)) console.log(x) # Scheme, 60 bytes (define(f n)(cond((= 1 n)#t)((<(abs n)1)#f)(#t(f(/ n -2)))))  Recursive solution. Axiom, 50 bytes g(n:INT):INT==(n=0 or n=1=>n;n rem 2=0=>g(n/-2);0)  It would return 0 if n is not power of (-2), else return 1; exercises (70) -> n:=-10000;repeat(if g(n)=1 then output n; n>10000=>break;n:=n+1) - 8192 - 2048 - 512 - 128 - 32 - 8 - 2 1 4 16 64 256 1024 4096  # Excel, 28 bytes =(-2)^INT(LOG(ABS(B2),2))=B2  Or using approach from @Martin Ender's Mathematica approach: 29 bytes =ISEVEN(LOG(MAX(B1,-2*B1),2))  # C#, 104 107 bytes +3 bytes, for using system, and finding another method to count bits using System;b=>Enumerable.Range(0,1+(int)Math.Log(int.MaxValue,2)).Select(x=>Math.Pow(-2,x)).Any(x=>b==x);  It uses Linq to calculate all of -2-s integer powers, and then to test if the input is one them. It would be a bit shorter, if it didn't have to work in a theoretical version, where int can be of any size. Ungolfed: bool IsPowerOfMinus2(int number) { return Enumerable.Range(0, 2 + (int)Math.Log(int.MaxValue, 2)) // We generate an IEnumerable, with values from 0 to the length of // maximum number in binary. We need to add one, because we need to // know, how many integers to .Select(x => Math.Pow(-2, x)) //Replace every number with -2 raised to the number .Any(x => number == x); //Determine, if the input is one of them }  # Jelly, 6 bytes AḶ-2*i  Try it online! Explanation:  Argument: -8 A Get the absolute value of n 8 Ḷ Create a range from 0 to that number-1 0, 1, 2, 3, 4, 5, 6, 7 -2* converts that range into the list -2^0, -2^1, ..., -2^7 i Returns >0 if n is in this range, 0 otherwise.  Dennis pointed out a flaw in my approach, which I fixed by taking the absolute value of the input for the range generation. # Pxem, 53 bytes (filename). Been a while since my last post of Pxem. Backslash followed by three digits of octet: a character whose codepoint is so: ._.c\001.y\001.r.-\002.!XX.a.c\003.x.c\004.%.w.d.a\004.$.c\003.a\001.z.d.aY.o


Try it online! Through this problem I noticed a bug in my interpreter. Maybe I need to redesign the interpreter; shortening the code makes maintainance difficult.

## Usage

• As a string of decimal integer from stdin for input.
• Outputs a letter Y for truthy; nothing for falsey.

## How it works: with comments

XX.z
# push an integer of input
.a._XX.z
# NOTE Pxem does not have negative constants
# So negative inputs must be changed to positive
# and then get multiplied by two
.a.c\001.y\001.r.-\002.!XX.aXX.z
# Pxem does not have log() nor something similar
# Just keep dividing by four
# And exit if it seemed not to be 4^n
.a.c\003.xXX.z
.a.c\004.%.w.d.aXX.z
.a\004.\$XX.z
.a.c\003.aXX.z
# final check for <4
.a\001.z.d.aY.o


# Brachylog, 10 bytes

1|¬0&~×₂ṅ↰


Try it online!

### Explanation

A recursive solution:

1|          Either the input is 1 (base case), or...
¬0        The input is not 0
&       and
~×₂    some integer times 2 equals the input
ṅ   Negate that integer
↰  and call the predicate recursively


# Japt-d, 7 bytes

Nø4ÍpUa


Try it