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Find a counterexample to show that each of the statements is false.

Find a counterexample to show that each of the statements is false.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Find a counterexample to show that each of the statements is false.

**Statement (d):** Every positive integer can be expressed as the sum of the squares of two integers. This statement is an assertion, which is examined in mathematics as one of several results regarding the representation of numbers as sums of squares. However, it is important to note that not every positive integer meets this condition. For example, integers that satisfy the condition "4k + 3" (where k is an integer) cannot be expressed as the sum of two squares. A classic example of such a number is 3, which cannot be expressed as the sum of the squares of two integers. This statement relates to an area in number theory focused on determining which numbers can be expressed as sums of squares, and understanding the conditions under which such representations are possible.
Transcribed Image Text:**Statement (d):** Every positive integer can be expressed as the sum of the squares of two integers. This statement is an assertion, which is examined in mathematics as one of several results regarding the representation of numbers as sums of squares. However, it is important to note that not every positive integer meets this condition. For example, integers that satisfy the condition "4k + 3" (where k is an integer) cannot be expressed as the sum of two squares. A classic example of such a number is 3, which cannot be expressed as the sum of the squares of two integers. This statement relates to an area in number theory focused on determining which numbers can be expressed as sums of squares, and understanding the conditions under which such representations are possible.
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