[vturvens] factoring round(2^2^decimal)

we consider rounded doubly exponential decimal powers of 2, the exponent having one digit past its decimal point.

the most difficult to factor were 7.8, 8.7, 8.8, and 8.9 , but we generally had good luck, getting all the way to 549 bits with Pari/GP.

the next unfactored term, round(2^2^9.2)/2^2/587/51307519 , is 552 bits or 166 digits.

factorint(round(2^2^0.0)) = Mat([2, 1])
factorint(round(2^2^0.1)) = Mat([2, 1])
factorint(round(2^2^0.2)) = Mat([2, 1])
factorint(round(2^2^0.3)) = Mat([2, 1])
factorint(round(2^2^0.4)) = Mat([2, 1])
factorint(round(2^2^0.5)) = Mat([3, 1])
factorint(round(2^2^0.6)) = Mat([3, 1])
factorint(round(2^2^0.7)) = Mat([3, 1])
factorint(round(2^2^0.8)) = Mat([3, 1])
factorint(round(2^2^0.9)) = Mat([2, 2])
factorint(round(2^2^1.0)) = Mat([2, 2])
factorint(round(2^2^1.1)) = Mat([2, 2])
factorint(round(2^2^1.2)) = Mat([5, 1])
factorint(round(2^2^1.3)) = [2, 1; 3, 1]
factorint(round(2^2^1.4)) = [2, 1; 3, 1]
factorint(round(2^2^1.5)) = Mat([7, 1])
factorint(round(2^2^1.6)) = Mat([2, 3])
factorint(round(2^2^1.7)) = [2, 1; 5, 1]
factorint(round(2^2^1.8)) = Mat([11, 1])
factorint(round(2^2^1.9)) = Mat([13, 1])
factorint(round(2^2^2.0)) = Mat([2, 4])
factorint(round(2^2^2.1)) = [2, 2; 5, 1]
factorint(round(2^2^2.2)) = [2, 3; 3, 1]
factorint(round(2^2^2.3)) = [2, 1; 3, 1; 5, 1]
factorint(round(2^2^2.4)) = [3, 1; 13, 1]
factorint(round(2^2^2.5)) = [2, 1; 5, 2]
factorint(round(2^2^2.6)) = Mat([67, 1])
factorint(round(2^2^2.7)) = [2, 1; 3, 2; 5, 1]
factorint(round(2^2^2.8)) = Mat([5, 3])
factorint(round(2^2^2.9)) = [3, 1; 59, 1]
factorint(round(2^2^3.0)) = Mat([2, 8])
factorint(round(2^2^3.1)) = [3, 1; 127, 1]
factorint(round(2^2^3.2)) = [2, 3; 73, 1]
factorint(round(2^2^3.3)) = [2, 1; 461, 1]
factorint(round(2^2^3.4)) = [2, 1; 3, 1; 251, 1]
factorint(round(2^2^3.5)) = [5, 1; 509, 1]
factorint(round(2^2^3.6)) = [41, 1; 109, 1]
factorint(round(2^2^3.7)) = [2, 1; 5, 1; 19, 1; 43, 1]
factorint(round(2^2^3.8)) = [5, 1; 3119, 1]
factorint(round(2^2^3.9)) = [2, 4; 1949, 1]
factorint(round(2^2^4.0)) = Mat([2, 16])
factorint(round(2^2^4.1)) = [2, 1; 5, 1; 73, 1; 199, 1]
factorint(round(2^2^4.2)) = [2, 4; 3, 1; 7103, 1]
factorint(round(2^2^4.3)) = [2, 3; 43, 1; 2473, 1]
factorint(round(2^2^4.4)) = [2, 2; 3, 1; 17, 1; 41, 1; 271, 1]
factorint(round(2^2^4.5)) = [7, 1; 925621, 1]
factorint(round(2^2^4.6)) = [2, 4; 1248241, 1]
factorint(round(2^2^4.7)) = [7, 1; 13, 2; 56417, 1]
factorint(round(2^2^4.8)) = [5, 1; 48641687, 1]
factorint(round(2^2^4.9)) = [1901, 1; 511549, 1]
factorint(round(2^2^5.0)) = Mat([2, 32])
factorint(round(2^2^5.1)) = [3, 1; 57859, 1; 121579, 1]
factorint(round(2^2^5.2)) = [2, 3; 3, 1; 7, 1; 13, 2; 31, 1; 132071, 1]
factorint(round(2^2^5.3)) = [2, 1; 5, 1; 72371023919, 1]
factorint(round(2^2^5.4)) = [14249, 1; 360563993, 1]
factorint(round(2^2^5.5)) = [3, 1; 274139, 1; 51047021, 1]
factorint(round(2^2^5.6)) = [2, 2; 3, 1; 1451, 1; 22908053449, 1]
factorint(round(2^2^5.7)) = [2, 1; 3, 1; 5, 1; 7, 2; 13, 2; 269, 1; 66655063, 1]
factorint(round(2^2^5.8)) = [2, 5; 131, 1; 274667, 1; 51372359, 1]
factorint(round(2^2^5.9)) = [2767, 1; 341766549877549, 1]
factorint(round(2^2^6.0)) = Mat([2, 64])
factorint(round(2^2^6.1)) = [1584509, 1; 281065007595277, 1]
factorint(round(2^2^6.2)) = [2, 7; 5, 1; 7, 1; 569, 1; 7757, 1; 19571, 1; 34916701, 1]
factorint(round(2^2^6.3)) = [7, 1; 103, 1; 127, 1; 42195053, 1; 135559173377, 1]
factorint(round(2^2^6.4)) = [3, 1; 5, 2; 409, 1; 1129, 1; 2870801, 1; 265492269499, 1]
factorint(round(2^2^6.5)) = Mat([1762483107300123635910219391, 1])
factorint(round(2^2^6.6)) = [32145331968121, 1; 4949436734577919, 1]
factorint(round(2^2^6.7)) = [2, 1; 97, 1; 151, 1; 556817, 1; 1216431791352251253211, 1]
factorint(round(2^2^6.8)) = [2, 3; 1585594054433, 1; 275824308477556951283, 1]
factorint(round(2^2^6.9)) = [13, 1; 53, 1; 10973, 1; 118285858337632191114889521413, 1]
factorint(round(2^2^7.0)) = Mat([2, 128])
factorint(round(2^2^7.1)) = [2, 1; 7, 2; 83, 1; 9041, 1; 38749, 1; 91249, 1; 50444291, 1; 15121041571170791, 1]
factorint(round(2^2^7.2)) = [2, 3; 17, 1; 1342522821832790664153757726489105358813887, 1]
factorint(round(2^2^7.3)) = [3, 1; 109, 1; 2699, 1; 2096147, 1; 17074529, 1; 8684358893347316238495921461, 1]
factorint(round(2^2^7.4)) = [2, 2; 89, 1; 563, 1; 3476230546241432207187585408972809840628116641, 1]
factorint(round(2^2^7.5)) = [2, 1; 5, 1; 17, 1; 29, 1; 83, 1; 18461, 1; 393637, 1; 4871897, 1; 3049600249, 1; 70312536722947040023493, 1]
factorint(round(2^2^7.6)) = [7, 1; 307, 1; 821, 1; 4409411, 1; 11627321, 1; 15910621369, 1; 1814613067213, 1; 9692513040339407, 1]
factorint(round(2^2^7.7)) = [3, 1; 7, 1; 9070337, 1; 2066879379621471258717712480808106605089939436363950741, 1]
factorint(round(2^2^7.8)) = [2, 2; 17253464873936579941191397, 1; 177375140304963462470398842141356922976613, 1]
factorint(round(2^2^7.9)) = [17, 1; 103, 1; 148931, 1; 2473411751, 1; 1668765871246896641255423, 1; 743005062591314795012392433209, 1]
factorint(round(2^2^8.0)) = Mat([2, 256])
factorint(round(2^2^8.1)) = [7, 1; 3407, 1; 26093335601, 1; 163543045324543, 1; 386521899947232358991722969819512144361364061724911949, 1]
factorint(round(2^2^8.2)) = [3, 2; 67, 1; 83, 1; 8737, 1; 76236584432059588836201231742718433160517232370902831181381530749315461827232631, 1]
factorint(round(2^2^8.3)) = [2, 4; 3, 1; 89, 1; 16151355697, 1; 11998345536743, 1; 225795199773133, 1; 402570031657967444662743845307575820055156242346316587, 1]
factorint(round(2^2^8.4)) = [2, 2; 3, 1; 5, 1; 7, 2; 881, 1; 3853, 1; 72719, 1; 101267, 1; 26487178889, 1; 19111595290223, 1; 1014222609575385221951, 1; 12865595880075367782274027157959366799, 1]
factorint(round(2^2^8.5)) = [2, 7; 5, 1; 2417, 1; 2777, 1; 57193, 1; 150594219991, 1; 1100795632146869, 1; 3527429957747951, 1; 67166245237612191749620350262962272512486476774464567, 1]
factorint(round(2^2^8.6)) = [2, 1; 41, 1; 10654155331857599, 1; 11439583978269157, 1; 64113821286468053696440099023378000874024254995105476162807950488867608546762993861, 1]
factorint(round(2^2^8.7)) = [3, 1; 101, 1; 9337, 1; 10799, 1; 10089487, 1; 10091723, 1; 63387027153443713, 1; 195720952888196462524736632595722803209, 1; 4016151252960955037314868658426392274250001347, 1]
factorint(round(2^2^8.8)) = [3, 1; 7, 1; 135559, 1; 247852121737012619907940371581, 1; 212382066624872762975584443464618314183107953999820671209368705106131941195407879042279845296005043, 1]
factorint(round(2^2^8.9)) = [2, 4; 419, 1; 247064781130769418452736004301838367529, 1; 386157843023199137206089018012232740550203329976777031980769740814780990183121328858674662573108149133, 1]
factorint(round(2^2^9.0)) = Mat([2, 512])
factorint(round(2^2^9.1)) = [2, 1; 3, 2; 13511888393, 1; 25359133111235715150527879, 1; 250893240957685558577690284467747048430450930670504192639959427274992229642543367368095758731596041005767735033443385319806310041, 1]