About a year ago you were asked to find the XOR primes. These are numbers whose only factors are 1 and themselves when performing XOR multiplication in base 2. Now were are going to spice things up a bit.

We are going to find the XOR primes in base -2

Converting to Base -2

Base -2 is a lot like every other base. The left most place is the 1s place (1 = (-2)0), next to that its the -2s place (-2 = (-2)1), next to that is the 4s place (4 = (-2)2), and so on and so forth. The big difference is that negative numbers can be represented in base -2 without any negative sign.

Here are some example conversions:

Decimal | Base -2
 6      |   11010
-7      |    1001
 12     |   11100
-15     |  110001

XOR addition in Base -2

XOR addition in Base -2 is pretty much the same as XOR addition in binary. You simply convert the number to Base -2 and XOR each digit in place. (This is the same as addition without the carry)

Here is an example worked through step by step:

(We will use the symbol +' to indicate Base -2 XOR addition)

Start in base 10:

6 +' -19

Convert to base -2:

11010 +' 10111

Add them without carrying:

+' 10111

Convert your result back into base 10:


XOR multiplication in Base -2

Once again XOR multiplication in base -2 is nearly the same as XOR multiplication in binary. If you are not familiar with XOR multiplication in base 2 there is an excellent explanation here I suggest you take a look at that first.

XOR multiplication in Base -2 is the same as performing long multiplication in base -2 except when it comes to the last step instead of adding up all of the numbers with a traditional + you use the +' we defined above.

Here is an example worked out below:

Start in decimal:

8 *' 7

Convert to Base -2:

11000 *' 11011

Set up long division:

*' 11011

Multiply the first number by every place in the second

*'    11011

Add up all the results using base -2 XOR addition

*'     11011
+' 11000

Convert the result back to decimal:


The challenge

Your challenge is to verify whether or not a number is an XOR prime in base -2. A number is an XOR prime in base -2 if the only pair of integers that multiply to it in base are 1 and itself. (1 is not prime)

You will take in a number and output a boolean, truthy if the input is an XOR prime in base -2 falsy otherwise.

Solutions will be scored in bytes with attaining the lowest number of bytes as the goal.

Test cases

The following are all XOR primes in base -2:


The following are not XOR primes in base -2:

  • \$\begingroup\$ 258 seems to equal -2 *' -129 = 10 *' 10000011 \$\endgroup\$ – JungHwan Min Jan 14 '17 at 3:46
  • \$\begingroup\$ @JungHwanMin my bad that one was supposed to be in the other category. I apologize if this has caused you any trouble. \$\endgroup\$ – Sriotchilism O'Zaic Jan 14 '17 at 3:58

Mathematica, 156 101 bytes


As stated here, this works because XOR multiplication is essentially multiplication in the polynomial ring F_2.



Start with {input}. Repeatedly replace a number a (except 0 and 1) by a mod 2 and prepend -floor(a/2), until it does not change. This calculates the input in base -2.

FromDigits[ ... ,x]

Create a polynomial using the digits of the base -2 number, using x as the variable. e.g. {1, 1, 0} -> x^2 + x

IrreduciblePolynomialQ[ ... ,Modulus->2]

Check whether the resulting polynomial is irreducible, with modulus 2.

Old version (156 bytes)


List of primes

Here's a list of base -2 XOR primes between -1000 and 1000 (pastebin)


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