# Even number not the sum of two base $2$ palindromes

What's the least positive even number not a sum of two base $$2$$ palindromes? I've checked and it must be over $$100$$ since all up to $$100$$ are such sums. [Or, which to me seems unlikely, are all even numbers such sums?]

Base $$2$$ palindromes entry o.e.i.s. http://oeis.org/A006995

• leading $0$'s not allowed, I presume? – Anvit Dec 27 '18 at 17:39
• @Anvit Yes, not with leading $0'$s -- all palindromes are thus odd numbers. – coffeemath Dec 27 '18 at 17:49

$$176$$ is your first culprit. Here's the Pseudocode (Python) I used. List was taken from OEIS.

from itertools import product

mylist = [0,1,3,5,7,9,15,17,21,27,31,33,45,51,63,65,73,85,
93,99,107,119,127,129,153,165,189,195,219,231,255,
257,273,297,313,325,341,365,381,387,403,427,443,
455,471,495,511,513,561,585,633,645,693,717,765,
771,819,843]

valids = set()
for i,j in product(mylist,mylist):

for i in range(0,850,2):
if i not in valids:
print(i)
break

• Thanks! I don't have (or know how to program) much knowledge on python or other CAS which would allow me to check. – coffeemath Dec 27 '18 at 17:51
• @coffeemath You can check 176 by hand aswell since you do have a list of binary palindromes :P – Anvit Dec 27 '18 at 17:53
• @coffeemath or, in any computer language, you can download the first ten thousand palindromes to a text file from oeis.org/A006995/b006995.txt and then write a program to add them in pairs, finally sort them and report those even numbers not achieved up to some bound. In C++ I can make a set type, it will automatically keep the pairwise sums in sorted order and not keep duplicates – Will Jagy Dec 27 '18 at 18:56