perplexus.info :: Numbers : Checking the quantity

Checking the quantity (Posted on 2023-09-23)

p^a+q^b=r^c

How many distinct solutions of the equation above are there, subject to the following constraints:

p, q, & r distinct primes
a, b, & c distinct positive integers,
each more than one
None of the powers exceeds 1111.

No Solution Yet Submitted by Ady TZIDON

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Puzzle Answer

| Comment 3 of 8 |

[2,4, 3, 2, 5, 2]

[2, 5, 7, 2, 3, 4]

[2, 7, 17, 3, 71, 2]

[7, 3, 13, 2, 2, 9]

2^4+3^2 =5^2 -->a, b, and c are NOT distinct. So, this is ignored.

2^5+7^2 = 3^4

2^7+17^3 =71^2 --> 5041 exceeds 1111. So, this is ignored.

7^3+13^2 = 2^9

Therefore, there are only 2 valid solutions, and these are:

(p, a, q,b,r,c) = (2,5, 7,2,3,4), (7,3,13,2,2,9)

Edited on September 23, 2023, 9:03 pm

Posted by K Sengupta on 2023-09-23 21:02:27

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