Find numbers that are palindromic in consecutive number bases

Question

I want to generate numbers which are palindromic in three or more consecutive number bases in the most optimal, fastest way (up to some range). I do not count trivial one digit palindromes.

(When I say 3 or more, I mean 3 and 4, as it is not known if a solution for 4 or more bases exists)

I'm basically generating palindromes in number base \$b\$, and then converting and checking whether it is also palindromic in \$b+1, b+2, \dots\$

Are there any ways to noticeably speed up my code?

---

# Converts any number n to any base b (*), Source: [1]
# https://stackoverflow.com/a/28666223/5821790
def numberToBase(n, b):
    if n == 0:
        return [0]
    digits = []
    while n:
        digits.append(int(n % b))
        n //= b
    return digits[::-1]

# Generates palindromes in base b (*), Source: [2]
# https://math.stackexchange.com/q/2494909
def palgen(b):
    i = 1
    while True:
        ii = b * i
        r = range(i, ii)
        for j in r:
            s = numberToBase(j, b)
            yield s + s[-2::-1]
        for j in r:
            s = numberToBase(j, b)
            yield s + s[::-1]
        i = ii

# Checks if the list is palindromic, Source: [3]
# https://stackoverflow.com/a/30340347/5821790
def isPalindrome(s):
    if len(s) <= 1:
        return True
    return s[0] == s[-1] and isPalindrome(s[1:-1])

# converts number in base b (*) to integer
def listToInt(digitList, base):
    l = len(digitList)
    value = 0
    for i, val in enumerate(digitList):
        value += val*base**(l-i-1)
    return value

# returns current time
def getTime():
    return strftime("( %H:%M:%S )", gmtime())


###################################################################
# Searches for numbers palindromic in 3 or more consecutive bases #
###################################################################

from time import gmtime, strftime
from math import sqrt, floor

bound = 10**8                       # numbers up to
baseBound = floor(sqrt(bound))      # bases up to (bound, can be improved)
print(getTime(), "Starting with:" ,baseBound, bound)

for b in range(2, baseBound):
    for i, s in enumerate(palgen(b), 1):

        # convert palindrome s_b to integer x and check if out of bound
        x = listToInt(s, b)
        if (x > bound): break

        if (len(s) > 1): # one digit palindromes do not count (trivial)

            # checks if the palindrome x is also palindromic in more bases
            if (isPalindrome(numberToBase(x, b+1))):
                if (isPalindrome(numberToBase(x, b+2))):
                    print(getTime(), b, x, len(s))

                    if (isPalindrome(numberToBase(x, b+3))):
                        print(b, x, len(s), "*** AT LEAST FOUR IN A ROW ***")

What are some things here that can be improved, and how, following good practice?

(Beside mathematical aspects which include the bound above which no more examples are found, the fact that only odd digit length palindromes form consecutive triples, and the facts that some examples follow a pattern that can be generated.)

Sources: [1] [2] [3]

Outputs: [10^9, ~ 3 hours: 1200 bases] and [10^12, ~ 3 hours: 100 bases]

Answer 1

A small performance improvement is possible in the listToInt function, by using multiplications only instead of exponentiation (which is Horner's method):

# converts number in base b list representation to integer
def listToInt(digitList, base):
    value = 0
    for val in digitList:
        value = value * base + val
    return value

On my computer this reduced the time to compute the results for \$ b = 1270, \ldots, 1286 \$ from 130 to 101 seconds.

Answer 2

This numberToBase function can be defined as: def numberToBase(n, b): return int(str(n), b)) Also as stated in comment isPalindrom can be defined as: def isPalindrom(s): return s == s[::-1] This may speed it up a bit.