It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 15 397 763 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 15 397 763, the only two divisors are 1 and 15 397 763. Therefore 15 397 763 is a prime number.
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Since 15397763 is a prime number, 15397763 is also a deficient number, that is to say 15397763 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 15397763 without 15397763 itself (that is 1, by definition!).