It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 573 506 707 is a prime number.
Indeed, the definition of a prime numbers is to have exactly two distinct positive divisors, 1 and itself. A number is a divisor of another number when the remainder of Euclid’s division of the second one by the first one is zero. Concerning the number 573 506 707, the only two divisors are 1 and 573 506 707. Therefore 573 506 707 is a prime number.
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Since 573506707 is a prime number, 573506707 is also a deficient number, that is to say 573506707 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 573506707 without 573506707 itself (that is 1, by definition!).