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 2020-07-14, 15:51 #7 kriesel     "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 10010111100112 Posts 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.