It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 346 529 137 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 346 529 137, the only two divisors are 1 and 346 529 137. Therefore 346 529 137 is a prime number.
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Since 346529137 is a prime number, 346529137 is also a deficient number, that is to say 346529137 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 346529137 without 346529137 itself (that is 1, by definition!).