Prime hunt using Beal's Conjecture

- Ramanraj K

8th June, 2020

Beal's Conjecture states that if Ax + By = Cz, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. The equation in the Conjecture is derived directly from the identity Ax + (A-1)Ax = A(x+1) where A=2, and in all other cases where A>2, may be derived directly or by repeated substitution. It may also be noted that A, B and C must have a common prime factor or a set of common prime factors, and the greatest prime factor must be common. Further, at least one of the terms A, B and C would have either 2 or 3 as a prime factor.This property may be used to hunt for primes.

Let x=3, y=3 and z=4.

To compute A, B and C, enter two prime numbers greater than 3 and submit:

Prime number #1:

Prime number #2:

Ax + By = Cz
Color Term #1: Color Term #2:
Details
Details
Details

Product of two given primes = 5 * 7 = 35

The coefficient of the second term would then be (35 - 1) = 34

The second term would then be 343 + 1 = 39304 + 1 = 39305

B3 = 13363703 = 2386602839305853000

Now, substituting the value of B in the identity Ax + (A-1)Ax = A(x+1) we get:

393053 + 13363703 = 393054 = 60721627297625 + 2386602839305853000 = 2386663560933150625

At this point it is already known that the first term would have at least one prime greater than the two given primes, and additionally, 2 or 3 would be one of the prime factors in one of the terms.

Common Prime factors of 60721627297625, 2386602839305853000, 2386663560933150625: 5 5 5 7 7 7 1123 1123 1123 = 53 73 11233

Greatest Common Prime factor of 60721627297625, 2386602839305853000, 2386663560933150625: 1123

As it can be seen, a prime number greater than the given primes was found. It would be possible to find it through direct division or in a few steps.

Note: However, if the given Prime Numbers are greater than 5 and 7, it goes beyond integer limits on most systems.

TERM #1: 393053 = 60721627297625
SCALE 1:39305 Length: 1 Width: 1 Height: 1

TERM #2: 13363703 = 2386602839305853000
SCALE 1:39305 Length: 39304 Width: 1 Height: 1

TERM #3: 393054 = 2386663560933150625
SCALE 1:39305 Length: 39305 Width: 1 Height: 1

Note: If the length, width or height is too large, it may not be rendered correctly.