Is there a number that is palindromal in both base 2 and base 3? I manually checked the first $20$ base $2$ palindromes and I did not find any base $3$ palindromes among them.
Is there any definite way of determining this? What about other bases?
 A: Note that what you are looking for is basically the sequence $\text{A0607092}$ on OEIS. 
However, examining the comments on the OEIS page, it seems unlikely that we will find a closed form for all such $n$ that satisfies the condition. Also, it seems that such $n$ grows very quickly. You probably should just write a code. 
A collections of code can be seen here. For example, you can code this on Python by  
from itertools import islice

digits = "0123456789abcdefghijklmnopqrstuvwxyz"

def baseN(num,b):
  if num == 0: return "0"
  result = ""
  while num != 0:
    num, d = divmod(num, b)
    result += digits[d]
  return result[::-1] # reverse

def pal2(num):
    if num == 0 or num == 1: return True
    based = bin(num)[2:]
    return based == based[::-1]

def pal_23():
    yield 0
    yield 1
    n = 1
    while True:
        n += 1
        b = baseN(n, 3)
        revb = b[::-1]
        #if len(b) > 12: break
        for trial in ('{0}{1}'.format(b, revb), '{0}0{1}'.format(b, revb),
                      '{0}1{1}'.format(b, revb), '{0}2{1}'.format(b, revb)):
            t = int(trial, 3)
            if pal2(t):
                yield t

for pal23 in islice(pal_23(), 6):
    print(pal23, baseN(pal23, 3), baseN(pal23, 2))

