Many people are aware of being a number between 3 and 4, and some also know that is between 2 and 3. Although the difference is less than 1/2, it’s enough to place the two constants in separate buckets on the number line, separated by an integer.
When dealing with powers of , using is frequently wasteful, so it helps to know that . Similarly, is way more precise than . To summarize: is between 7 and 8, while is between 9 and 10.
Do any two powers of and have the same integer part? That is, does the equation have a solution in positive integers ?
Probably not. Chances are that the only pairs for which are , the smallest difference attained by .
Indeed, having implies that , or put differently, . This would be an extraordinary rational approximation… for example, with it would mean that with the following digits all being . This isn’t happening.
Looking at the continued fraction expansion of shows the denominators of modest size , indicating the lack of extraordinarily nice rational approximations. Of course, can use them to get good approximations, , which leads to with small relative error. For example, dropping and subsequent terms yields the convergent , and one can check that while .
Trying a few not-too-obscure constants with the help of mpmath library, the best coincidence of integer parts that I found is the following: the 13th power of the golden ratio and the 34th power of Apèry’s constant both have integer part 521.