The 13th root of 9814843037474615665221286141505078084763648 is an integer.

Without direct evaluation, determine the foregoing integer, using only simple calculator and p&p.

The number has 43 digits

X is 43 digit number betw 10^42 and 10^43

n^13 = X
13 * log13(n) = log13(x)
log13(n) = log13(x)/13

log13(10^42)/13 = 2.9002993950084743 = a
log13(10^43)/13 = 2.969354142508676 = b

13^a = 1701.3
13^b = 2030.9

for 1702  up to 2030, raising to 13 gives 43 digits:  329 possibilities
Since this 13th power ends in 8, the base number also ends in 8:  33 possiblities
Since the first 2 digits are 98, it is much closer to 2028 than it is to 1708.
In fact since the number is in the top 2% of 43 digit numbers and we are down to only 33 possibilities, I bet a dollar that our number is 2028 and I bet 5 dollars that it is either 2028 or 2018, the first of which is divisible by 3.  From parsing through the 43 digits, we see that this number is also divisible by 3.

Final Answer:  2028^13 = our 43 digit number which calculator confirms.

Posted by Larry on 2023-10-11 15:41:26