It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 39 702 629 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 39 702 629, the only two divisors are 1 and 39 702 629. Therefore 39 702 629 is a prime number.
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Since 39702629 is a prime number, 39702629 is also a deficient number, that is to say 39702629 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 39702629 without 39702629 itself (that is 1, by definition!).