It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 37 589 131 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 37 589 131, the only two divisors are 1 and 37 589 131. Therefore 37 589 131 is a prime number.
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Since 37589131 is a prime number, 37589131 is also a deficient number, that is to say 37589131 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 37589131 without 37589131 itself (that is 1, by definition!).