Q: Counter example that a positive integer is not equal to the sum of two squares?

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The proposition in the question is simply not true so there can be no answer!For example, if given the integer 6:there are no two perfect squares whose sum is 6,there are no two perfect squares whose difference is 6,there are no two perfect squares whose product is 6,there are no two perfect squares whose quotient is 6.

factors are 2 x 36, 4 x 18 and 8 x 9 so there are three perfect squares, 4, 9 and 36.

This is the series of integer squares, also known as perfect squares. It is one example of a "power law" series.

The positive integer factors of 36 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36 The perfect squares in this list are: 1, 4, 9, 36

perfect squares

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#include<iostream> #include<sstream> using namespace std; unsigned sum_of_squares (const unsigned max) { if (max==0) return 0; if (max==1) return 1; return sum_of_squares (max-1) + (max*max); } int main () { unsigned num = 0; while (1) { cout << "Enter a positive integer (0 to exit): "; string s; cin >> s; if (s[0]=='0') break; stringstream ss; ss << s; if (ss >> num) { cout << "The sum of all squares from 1 to " << num << " is: " << sum_of_squares (num) << endl; continue; } cerr << "Invalid input: " << s << endl; } cout << "Quitting..." << endl; } Example output: Enter a positive integer (0 to exit): 1 The sum of all squares from 1 to 1 is: 1 Enter a positive integer (0 to exit): 2 The sum of all squares from 1 to 2 is: 5 Enter a positive integer (0 to exit): 3 The sum of all squares from 1 to 3 is: 14 Enter a positive integer (0 to exit): 4 The sum of all squares from 1 to 4 is: 30 Enter a positive integer (0 to exit): 5 The sum of all squares from 1 to 5 is: 55 Enter a positive integer (0 to exit): 6 The sum of all squares from 1 to 6 is: 91 Enter a positive integer (0 to exit): 7 The sum of all squares from 1 to 7 is: 140 Enter a positive integer (0 to exit): 0 Quitting...

The proposition in the question is simply not true so there can be no answer!For example, if given the integer 6:there are no two perfect squares whose sum is 6,there are no two perfect squares whose difference is 6,there are no two perfect squares whose product is 6,there are no two perfect squares whose quotient is 6.

a positive integer A that, if increased or decreased by the same positive integer B, yields 2 positive integers, A+B and A-B, that are both perfect squares" OK... i figured out kinda what it meant... i think the integer B is equal to A-1, like the rectangular number definition: n(n-1)

It might seems like it, but actually no. Proof: sqrt(0) = 0 (0 is an integer, not a irrational number) sqrt(1) = 1 (1 is an integer, not irrational) sqrt(2) = irrational sqrt(3) = irrational sqrt(4) = 2 (integer) As you can see, there are more than 1 square root of a positive integer that yields an integer, not a irrational. While most of the sqrts give irrational numbers as answers, perfect squares will always give you an integer result. Note: 0 is not a positive integer. 0 is neither positive nor negative.

factors are 2 x 36, 4 x 18 and 8 x 9 so there are three perfect squares, 4, 9 and 36.

This is the series of integer squares, also known as perfect squares. It is one example of a "power law" series.

perfect squares

Prime squares have three factors. There are 11 of them in that range.

A perfect square is an integer (whole number) times itself. E.g. 3*3 = 9, or -4*-4 = 16. A negative number times a negative number is a positive number. This means a negative number times itself would be positive. It also holds true for all squares, not just perfect squares. E.g., -1.3 * -1.3 = 1.69 (which is positive).

The positive integer factors of 36 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36 The perfect squares in this list are: 1, 4, 9, 36

perfect squares

A rectangle with dimensions of 1" x 2" .