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Old 2020-07-14, 15:51   #7
kriesel
 
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"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest

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What formula?

The alleged prime 2^283243137 - 1 can easily be shown to be composite without using a computer or calculator.
Sum the decimal digits of the exponent: 33. (Sometimes called "casting out nines") The digit sum is obviously divisible by 3. Any number 2^n-1 where n= a x b is composite, is composite, and is a repdigit (number with repeating digits), with factors 2^a-1 and 2^b-1 easily visible when expressed in base 2^a and 2^b respectively.

Consider 2^8-1 = 255 = 2^(2*4)-1
2^2-1 = 3 = 255/85.
2^4-1 = 15 = 255/17.
2^4-1 = 15 = 2^2-1 * cofactor 5.

For an exponent with 4 distinct prime factors, for example from the OP, 283243137:
https://www.alpertron.com.ar/ECM.HTM: 283243137 = 3 × 17 × 23 × 241469
a=3 (repdigit 2^3 - 1 = 7's in base 2^3 = 8)
b=17 (repdigit 2^17 - 1 = 131071's in base 2^17 = 131072)
c=23 (repdigit 2^23 - 1 = 8388607's in base 2^23 = 8388608)
d=241469 (repdigit 2^241469 - 1 in base 2^241469)

The number has numerous factors (at least 14, as shown below), each of which corresponds to being able to express the number as a repdigit in some base 2^B where 2^B=factor+1. For an exponent with four distinct prime factors, a, b, c, d, there are unique factors as follows

prime factors
2^a-1
2^b-1
2^c-1
2^d-1

composite factors
2^(ab)-1
2^(ac)-1
2^(ad)-1
2^(bc)-1
2^(bd)-1
2^(cd)-1

2^(abc)-1
2^(abd)-1
2^(bcd)-1

There's also a cofactor, whatever 2^(abcd)-1 / (2^a-1) / (2^b-1) / (2^c-1) / (2^d-1) is. Which may be prime or composite.
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