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It is known that "if three prime numbers a, B and C greater than 3 satisfy the relation 2A + 5B = C, then a + B + C is a multiple of integer n". What is the maximum possible value of integer n in this theorem? Please prove your conclusion
This paper proves that: a + A + B + B + C = a + B + B + C = a + B + B + 2A + 5B = 3 (a + 2b), and obviously, 3|a + B + C, if the remainder of a and B after 3 is respectively RA and Rb, then RA ≠0, and Rb ≠ 0. If RA ≠ Rb, then RA = 2, Rb = 1, or RA = 1, or RA = 1, Rb = 2, then 2A + 5B = 2 (3m + 2) + 5 (3 (3m + 2) + 5 (3N + 5 (3N + 1) = 3 (2m + 5N + 5N + 3) (2m + 5N + 2m + 5N + 3) or 2A + 5B = 2 (2 (2) (3P + 2 (3P + 1) + + 1) + 2) (2 (3 (3P + 1) + 5 (3P) + 5) + 5) (5 (3P = 2) (2P only RA = RA = RA = RA = RA = 3) then ra= Then a + 2B must be a multiple of 3, so a + B + C is a multiple of 9. A and B are prime numbers greater than 3. According to the meaning of the topic, take a = 11, B = 5, then 2A + 5B = 2 × 11 + 5 × 5 = 47, a + B + C = 11 + 5 + 47 = 63, take a = 13, B = 7, then 2A + 5B = 2 × 13 + 5 × 7 = 61, a + B + C = 13 + 7 + 61 = 81, and (63, 81) = 9, so 9 is the maximum possible value
What primes can be made of 1.2.3
If you choose one number, it will be 3. If you choose two numbers, it will be 13, 31, 23
Three numbers are multiples of 3 anyway, so there are only four, which are 3, 13, 23 and 31
A is prime, B is positive integer, and 9 (2a + b) ^ 2 = 509 (4a + 511b)
Python code, a, B within 10000, only found a group of 251 and 7 > > > for a in xrange (210000): for B in xrange (110000): if 9 * (2 * a + b) * (2 * a + b) = = 509 * (4 * a + 511 * b): print a, B answer a = 251, B = 7
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