Show that the product of two of the numbers 651000 - 82001 + 3177, 791212 _ 9239

Question

Show that the product of two of the numbers 651000 - 82001 + 3177, 791212 _ 92399 + 22001, and 244493 - 58192 + 71777 is nonnegative. Is your proof constructive or nonconstructive? [Hint: Do not try to evaluate these numbers!]

Explanation / Answer

65^1000 ? 8^2001 + 3^177 > 0 compute decimal log of each term of the sum take integer part and add 1 the result is the quantity of digits of the number int(1000 log 65) + 1 = 1813 int(2001 log 8) + 1 = 1808 int (177 log 3) + 1 = 85 adding to and subtracting from to a number with 1813 digits numbers with less digits does not change the number of digits of the first number, which is also the number of digits of the result 1813 in a similar way 79^1212 ? 9^2399 + 2^2001 has 2300 digits like 79^1212 and 24^4493 ?5^8192 + 7^1777 6202, the same number of digits of 24^4493 they are all positive as well as their product

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