It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 25 010 003 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 25 010 003, the only two divisors are 1 and 25 010 003. Therefore 25 010 003 is a prime number.
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Since 25010003 is a prime number, 25010003 is also a deficient number, that is to say 25010003 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 25010003 without 25010003 itself (that is 1, by definition!).