Meaning: Carl Friedrich Gauss, a renowned mathematician, made this statement in his work "Disquisitiones Arithmeticae" published in 1801. In this quote, Gauss highlights the significance of the problem of distinguishing prime numbers from composite numbers and the process of resolving composite numbers into their prime factors in the field of arithmetic.

Prime numbers are integers greater than 1 that have no divisors other than 1 and themselves. They play a fundamental role in number theory and have applications in various fields such as cryptography, computer science, and mathematics. On the other hand, composite numbers are integers that have more than two divisors, meaning they can be factored into smaller integers. Understanding and being able to distinguish between prime and composite numbers is crucial in many mathematical and practical applications.

Gauss's emphasis on the importance of this problem underscores its significance in arithmetic. The ability to identify prime numbers and factorize composite numbers is not only a foundational concept in mathematics but also has practical implications. For example, prime numbers are essential in cryptography for creating secure communication channels, as they form the basis of algorithms used in encryption and decryption.

Furthermore, the process of resolving composite numbers into their prime factors is a fundamental aspect of number theory. Prime factorization, the decomposition of a composite number into a product of prime numbers, is a crucial tool in various mathematical algorithms and problem-solving techniques. It is used in diverse mathematical areas such as algebra, geometry, and number theory, making it a fundamental concept in mathematics.

Gauss's acknowledgment of the importance and usefulness of this problem also reflects its historical and ongoing relevance in the field of arithmetic. The study of prime numbers and their properties has fascinated mathematicians for centuries, with the exploration of prime numbers leading to the discovery of new mathematical concepts and the development of advanced mathematical theories.

Moreover, the ability to efficiently factorize composite numbers into their prime factors has practical implications in fields such as cryptography, integer factorization algorithms, and data security. As technology continues to advance, the significance of prime numbers and their factorization becomes increasingly relevant in ensuring the security and integrity of digital communication and information systems.

In conclusion, Carl Friedrich Gauss's quote highlights the enduring importance and usefulness of the problem of distinguishing prime numbers from composite numbers and resolving composite numbers into their prime factors in arithmetic. This fundamental problem not only underpins various mathematical concepts and theories but also has practical applications in cryptography, computer science, and other fields. Gauss's recognition of the significance of this problem underscores its historical and ongoing relevance in the realm of mathematics and its broader impact on modern society.