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Exactly $1000$ perfect squares between two consecutive cubes Therefore there are exactly $1000$ squares between the successive cubes $ (667^2)^3$ and $ (667^2+1)^3$, or between $444889^3$ and $444890^3$ Finally, we can verify all of this by using the command line utility bc: $ bc sqrt((667^2)^3) 296740963 sqrt((667^2+1)^3-1) 296741963 Cite edited 17 hours ago community wiki 5 revs R P A reflection on
algebra precalculus - Which is greater: $1000^ {1000}$ or $1001^ {999 . . . The way you're getting your bounds isn't a useful way to do things You've picked the two very smallest terms of the expression to add together; on the other end of the binomial expansion, you have terms like $999^ {1000}$, which swamp your bound by about 3000 orders of magnitude
algebra precalculus - Multiple-choice: sum of primes below $1000 . . . Given that there are $168$ primes below $1000$ Then the sum of all primes below 1000 is (a) $11555$ (b) $76127$ (c) $57298$ (d) $81722$ My attempt to solve it: We know that below $1000$ there are $167$ odd primes and 1 even prime (2), so the sum has to be odd, leaving only the first two numbers
combinatorics - Determine the number of odd binomial coefficients in . . . 4 Determine the number of odd binomial coefficients in the expansion of $ (x+y)^ {1000}$ Hint: The number of odd coefficients in any finite binomial expansion is a power of $2$ Is there a way to prove this without using something like Lucas's theorem or any other non-trivial result?