2689

2689 is a natural number that follows 2688 and precedes 2690. It is notable in elementary number theory for being a prime number and for its particular representations in different bases.

As a prime, 2689 is odd and has no divisors other than 1 and itself. Its square root is approximately 51.8, so primality can be checked by testing primes up to 47; none of these divide 2689, confirming its status as a prime number.

2689 has a representation as a sum of two squares: 2689 = 33^2 + 40^2. This aligns with the fact that primes congruent to 1 modulo 4 can be written as the sum of two squares; indeed, 2689 ≡ 1 (mod 4).

In different numeral systems, 2689 has distinct representations. In binary it is 101010000001, and in hexadecimal it is A81. In Roman numerals, 2689 is written as MMDCLXXXIX.

Other numerical properties include its digit sum, which is 2 + 6 + 8 + 9 = 25, giving a digital root of 7. While these digit-focused observations are common curiosities, they do not imply deeper structural significance beyond standard arithmetic properties.

Overall, 2689 is a prime number with a notable sum-of-squares representation and recognizable forms in several common numeral systems, illustrating typical features characteristic of primes congruent to 1 modulo 4.

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