A number theory problem by Trung Le
Number Theory
Level
2
For how many values of n all the three numbers 3n-4, 4n-5, 5n-3 are prime?
The answer is 1.
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1 solution
Note that n > 1 and the sum of the three numbers is an even number (12n - 12).
Consider 2 cases:
a) All 3 numbers are even: this is impossible because 2 is the only even prime number.
b) One of them is even (and equals 2), the other two are odd:
If 3n - 4 = 2, then n = 2
4n-5 is an odd number, it can not be 2
If 5n - 3 = 2, then n = 1, rejected.
So the only value of n so that all the three numbers 3n-4, 4n-5, 5n-3 are prime is 2.