Assuming the number is [tex]2^{144,000}[/tex], note that this number requires 144,001 digits in binary. On average, a single digit in base 60 requires about [tex]2^n=60\implies n\approx5.9069[/tex] binary digits. But obviously, we have to use a whole number of digits, so we round up to the next integer; in other words, given a number [tex]x[/tex] in base 60 with [tex]n[/tex] digits, we should expect [tex]x[/tex] in base 2 to have [tex]\lceil5.9069n\rceil[/tex].
So, if the "conversion rate" from base 60 to base 2 is about 5.9069, then the reverse rate would be [tex]\dfrac1{5.9069}\approx0.169294[/tex], i.e. the solution to [tex]60^n=2[/tex]. Again, we'd have to round up, so if [tex]x[/tex] in base 2 has [tex]n[/tex] digits, then [tex]x[/tex] in base 60 should have [tex]\lceil0.169294n\rceil[/tex] digits.
This means [tex]2^{144,000}[/tex] in base 60 would have [tex]\lceil0.169294\times144,000\rceil\approx\lceil24,378.3\rceil=24,379[/tex] digits.
