for a prime number P, compute the prime factorization of P-1, then recursively repeat with each of those primes. as part of the Lucas primality test, this recursive factorization serves as a deterministic proof of primality (Pratt certificate). (each prime also needs a witness, but witnesses are easy to find, e.g., randomly.)
we rewrite the previous compact Recursive Prime Predecessor Factorization Notation into a more conventional form. although it is generally more verbose, it avoids repeating the definition of common prime factors. prime factors are defined in order of decreasing size.
the first two examples below are Mersenne primes.
each certificate tells a complicated multilayered story in a language that is mostly incomprehensible. certain "words", small factors, get used frequently. in Mersenne primes, the first "sentence", the factorization of the number 2^p-2, often has many prime factors because of algebraic factorization. perhaps the story of 2^3217-1 can be completed someday.
2^2281 - 1 = p262; p262 = 1 + 2 * p1^2 * p2^2 * p3 * p4 * p5 * p6 * p10 * p12 * p17 * p34 * p38 * p42 * p44 * p51 * p58 * p65 * p70 * p85 * p86 * p98 * p111 * p134 * p141 * p143 * p150 * p156 * p157 * p158 * p159 * p161 * p165 * p178 * p179 * p185 * p187 * p208 * p209 * p216 * p217 * p219 * p220 * p224 * p229 * p234 * p240 * p241 * p242 * p243 * p244 * p245 * p248 * p249 * p254 * p256 * p258 * p261; p261 = 1 + 2^5 * p1 * p2 * p7 * p8 * p12 * p221 * p239 * p259 * p260; p260 = 1 + 2^3 * p3 * p250 * p255; p259 = 1 + 2^8 * p1 * p7 * p24 * p228 * p233 * p252; p258 = 1 + 2^4 * p1 * p2 * p7 * p29 * p38 * p73 * p257; p257 = 1 + 2^5 * p2^2 * p3 * p8 * p74 * p251; p256 = 1 + 2^4 * p1 * p2 * p7 * p253; p255 = 1 + 2 * p8 * p97 * p128 * p162 * p198 * p237; p254 = 1 + 2^7 * p1 * p2 * p7 * p8 * p37 * p247; p253 = 1 + 2^7 * p1 * p2 * p200 * p205 * p236; p252 = 1 + 2^2 * p8 * p26 * p169 * p246; p251 = 1 + 2 * p2 * p4 * p5 * p11 * p223 * p235; p250 = 1 + 2 * p1 * p4 * p6^2 * p8 * p140 * p215 * p218; p249 = 1 + 2^2 * p1 * p2 * p7 * p50 * p183 * p238; p248 = 1 + 2^4 * p1 * p7^2 * p68 * p168 * p201 * p210; p247 = 1 + 2^3 * p1^2 * p3^2 * p47 * p113 * p146 * p175 * p207; p246 = 1 + 2^2 * p1 * p9 * p15 * p46 * p114 * p227; p245 = 1 + 2^3 * p1 * p2 * p7 * p14 * p180 * p225; p244 = 1 + 2^3 * p1 * p2 * p3 * p5 * p7 * p108 * p110 * p139 * p155; p243 = 1 + 2 * p1 * p2 * p7 * p53 * p66 * p191 * p197; p242 = 1 + 2 * p1 * p2 * p7 * p142 * p222; p241 = 1 + 2^5 * p2^2 * p7 * p231; p240 = 1 + 2^2 * p1^2 * p2 * p7 * p104 * p152 * p167; p239 = 1 + 2 * p84 * p230; p238 = 1 + 2^3 * p1^3 * p2 * p164 * p211; p237 = 1 + 2 * p2 * p232; p236 = 1 + 2^4 * p25 * p226; p235 = 1 + 2^2 * p1 * p2 * p188 * p204; p234 = 1 + 2^3 * p1 * p2 * p7 * p166 * p193; p233 = 1 + 2^2 * p12 * p94 * p214; p232 = 1 + 2 * p1 * p11 * p26 * p87 * p199; p231 = 1 + 2 * p2 * p8 * p10 * p117 * p196; p230 = 1 + 2 * p92 * p151 * p170; p229 = 1 + 2^4 * p1 * p6 * p7 * p99 * p181; p228 = 1 + 2 * p1 * p21 * p129 * p190; p227 = 1 + 2 * p8 * p11 * p213; p226 = 1 + 2^2 * p1^3 * p35 * p206; p225 = 1 + 2 * p3 * p149 * p172; p224 = 1 + 2^10 * p1^4 * p7 * p159; p223 = 1 + 2^5 * p33 * p40 * p163; p222 = 1 + 2^2 * p1^2 * p76 * p194; p221 = 1 + 2^3 * p1 * p6 * p29 * p186; p220 = 1 + 2 * p1^2 * p2 * p7 * p203; p219 = 1 + 2^4 * p1 * p5 * p7 * p184; p218 = 1 + 2^2 * p212; p217 = 1 + 2^4 * p1 * p2 * p6 * p7 * p42 * p45; p216 = 1 + 2^6 * p1^3 * p2 * p4 * p7 * p96; p215 = 1 + 2 * p1 * p2^2 * p18 * p171; p214 = 1 + 2^6 * p1 * p6 * p182; p213 = 1 + 2^2 * p1 * p9 * p14 * p147; p212 = 1 + 2 * p1 * p16 * p88 * p90; p211 = 1 + 2 * p1^3 * p5 * p6 * p145; p210 = 1 + 2^4 * p1^2 * p47 * p127; p209 = 1 + 2 * p2^4 * p3 * p7 * p100; p208 = 1 + 2^4 * p1 * p7 * p160; p207 = 1 + 2 * p2 * p202; p206 = 1 + 2 * p1 * p2 * p3 * p174; p205 = 1 + 2^2 * p2 * p10 * p41 * p62; p204 = 1 + 2 * p1 * p32 * p131; p203 = 1 + 2^3 * p192; p202 = 1 + 2^2 * p1 * p7^2 * p12 * p21; p201 = 1 + 2^2 * p3 * p33 * p105; p200 = 1 + 2 * p1 * p189; p199 = 1 + 2 * p1 * p3^2 * p133; p198 = 1 + 2^3 * p2 * p6 * p121; p197 = 1 + 2 * p195; p196 = 1 + 2^3 * p176; p195 = 1 + 2^2 * p3 * p153; p194 = 1 + 2 * p1 * p177; p193 = 1 + 2^2 * p1 * p53 * p63; p192 = 1 + 2^6 * p4 * p14 * p18; p191 = 1 + 2 * p41 * p112; p190 = 1 + 2 * p1 * p2 * p144; p189 = 1 + 2^3 * p1^2 * p2^2 * p78; p188 = 1 + 2 * p50 * p93; p187 = 1 + 2 * p1 * p7 * p123; p186 = 1 + 2 * p67 * p75; p185 = 1 + 2 * p1 * p2 * p7 * p91; p184 = 1 + 2 * p1 * p154; p183 = 1 + 2 * p173; p182 = 1 + 2 * p3 * p34 * p52; p181 = 1 + 2^3 * p148; p180 = 1 + 2 * p3 * p5 * p8 * p29; p179 = 1 + 2^10 * p1^3 * p7; p178 = 1 + 2 * p1^3 * p3 * p7 * p20; p177 = 1 + 2 * p1^3 * p2 * p9 * p15; p176 = 1 + 2^4 * p10 * p72; p175 = 1 + 2^3 * p7 * p101; p174 = 1 + 2^3 * p1^2 * p2 * p81; p173 = 1 + 2^5 * p124; p172 = 1 + 2^3 * p1^4 * p5 * p13; p171 = 1 + 2^2 * p1 * p132; p170 = 1 + 2 * p3 * p5 * p6 * p23; p169 = 1 + 2 * p1 * p138; p168 = 1 + 2^2 * p1 * p3 * p106; p167 = 1 + 2^2 * p1 * p2 * p109; p166 = 1 + 2^2 * p1^2 * p5 * p7 * p9; p165 = 1 + 2^2 * p1 * p2 * p3 * p7 * p10; p164 = 1 + 2 * p1^2 * p5 * p77; p163 = 1 + 2^3 * p1 * p120; p162 = 1 + 2^2 * p2 * p12 * p44; p161 = 1 + 2^2 * p1 * p2 * p7 * p36; p160 = 1 + 2^2 * p137; p159 = 1 + 2 * p1^2 * p3 * p7 * p20; p158 = 1 + 2^3 * p1 * p7 * p52; p157 = 1 + 2^5 * p2 * p3^2 * p7; p156 = 1 + 2^2 * p1 * p2^2 * p7 * p8; p155 = 1 + 2^3 * p16 * p47; p154 = 1 + 2 * p2 * p125; p153 = 1 + 2^2 * p1^3 * p2 * p4 * p6; p152 = 1 + 2^3 * p1^2 * p3 * p40; p151 = 1 + 2^2 * p1^2 * p102; p150 = 1 + 2^4 * p1^3 * p4 * p7; p149 = 1 + 2 * p7 * p99; p148 = 1 + 2 * p8 * p89; p147 = 1 + 2 * p1 * p126; p146 = 1 + 2 * p136; p145 = 1 + 2 * p135; p144 = 1 + 2 * p1 * p122; p143 = 1 + 2^4 * p1 * p2 * p45; p142 = 1 + 2^2 * p5 * p83; p141 = 1 + 2^6 * p1^2 * p2 * p7; p140 = 1 + 2^2 * p10 * p57; p139 = 1 + 2 * p7^2 * p19; p138 = 1 + 2^3 * p115; p137 = 1 + 2 * p1 * p118; p136 = 1 + 2 * p130; p135 = 1 + 2 * p7 * p79; p134 = 1 + 2^3 * p1 * p7 * p19; p133 = 1 + 2^2 * p116; p132 = 1 + 2 * p1^2 * p3^2 * p9; p131 = 1 + 2^2 * p1 * p95; p130 = 1 + 2 * p2 * p5 * p35; p129 = 1 + 2 * p1 * p5^2 * p6; p128 = 1 + 2^2 * p17 * p18; p127 = 1 + 2 * p119; p126 = 1 + 2^2 * p107; p125 = 1 + 2 * p1 * p9 * p20; p124 = 1 + 2^3 * p2^2 * p16; p123 = 1 + 2 * p1^3 * p39; p122 = 1 + 2 * p2 * p80; p121 = 1 + 2 * p7 * p45; p120 = 1 + 2 * p1 * p6 * p22; p119 = 1 + 2 * p2 * p71; p118 = 1 + 2 * p2 * p69; p117 = 1 + 2^2 * p1 * p64; p116 = 1 + 2 * p1 * p82; p115 = 1 + 2 * p103; p114 = 1 + 2^2 * p1 * p59; p113 = 1 + 2 * p2 * p5 * p13; p112 = 1 + 2^2 * p1 * p55; p111 = 1 + 2^4 * p1 * p2 * p7; p110 = 1 + 2 * p1^2 * p5 * p7; p109 = 1 + 2^2 * p2 * p5 * p6; p108 = 1 + 2^3 * p2 * p23; p107 = 1 + 2 * p1 * p4 * p15; p106 = 1 + 2 * p1 * p7 * p9; p105 = 1 + 2^2 * p2 * p3 * p8; p104 = 1 + 2^5 * p2 * p7; p103 = 1 + 2^3 * p3 * p15; p102 = 1 + 2^6 * p13; p101 = 1 + 2 * p10 * p12; p100 = 1 + 2 * p2 * p4 * p8; p99 = 1 + 2^2 * p66; p98 = 1 + 2^3 * p1 * p2 * p7; p97 = 1 + 2 * p1 * p54; p96 = 1 + 2^2 * p3 * p21; p95 = 1 + 2 * p1 * p2 * p18; p94 = 1 + 2 * p3 * p4 * p5; p93 = 1 + 2 * p1 * p51; p92 = 1 + 2 * p1 * p2^2 * p5; p91 = 1 + 2^2 * p1 * p3 * p8; p90 = 1 + 2 * p1 * p49; p89 = 1 + 2^2 * p1 * p2 * p10; p88 = 1 + 2^2 * p1 * p33; p87 = 1 + 2 * p1 * p2 * p3^2; p86 = 1 + 2^3 * p1 * p2 * p4; p85 = 1 + 2^6 * p7; p84 = 1 + 2 * p1^2 * p2 * p5; p83 = 1 + 2^7 * p1^2; p82 = 1 + 2 * p2 * p27; p81 = 1 + 2 * p1 * p2^2 * p3; p80 = 1 + 2 * p1 * p37; p79 = 1 + 2 * p61; p78 = 1 + 2^2 * p4 * p8; p77 = 1 + 2^4 * p1^2 * p3; p76 = 1 + 2 * p60; p75 = 1 + 2 * p2 * p3 * p5; p74 = 1 + 2 * p56; p73 = 1 + 2^2 * p1^2 * p8; p72 = 1 + 2 * p1 * p31; p71 = 1 + 2 * p1^4 * p2; p70 = 1 + 2^3 * p2 * p7; p69 = 1 + 2^2 * p1 * p17; p68 = 1 + 2^2 * p1 * p16; p67 = 1 + 2^2 * p1^2 * p6; p66 = 1 + 2^6 * p1^2; p65 = 1 + 2 * p1 * p2 * p7; p64 = 1 + 2^3 * p19; p63 = 1 + 2 * p48; p62 = 1 + 2 * p1^2 * p9; p61 = 1 + 2^2 * p29; p60 = 1 + 2 * p2 * p3^2; p59 = 1 + 2 * p43; p58 = 1 + 2^3 * p1 * p7; p57 = 1 + 2^4 * p1^3; p56 = 1 + 2 * p4 * p7; p55 = 1 + 2^4 * p2^2; p54 = 1 + 2^2 * p1 * p10; p53 = 1 + 2 * p1 * p17; p52 = 1 + 2^5 * p4; p51 = 1 + 2 * p1 * p2 * p4; p50 = 1 + 2^2 * p21; p49 = 1 + 2 * p2 * p10; p48 = 1 + 2^3 * p2 * p3; p47 = 1 + 2 * p1^3 * p2; p46 = 1 + 2 * p30; p45 = 1 + 2^8; p44 = 1 + 2^4 * p1 * p2; p43 = 1 + 2 * p3 * p6; p42 = 1 + 2^2 * p1 * p7; p41 = 1 + 2 * p28; p40 = 1 + 2 * p1^2 * p4; p39 = 1 + 2^2 * p3^2; p38 = 1 + 2 * p2 * p7; p37 = 1 + 2^2 * p13; p36 = 1 + 2 * p1^4; p35 = 1 + 2^2 * p1 * p5; p34 = 1 + 2 * p1 * p2^2; p33 = 1 + 2^2 * p11; p32 = 1 + 2 * p1 * p8; p31 = 1 + 2^3 * p6; p30 = 1 + 2 * p2 * p5; p29 = 1 + 2 * p1^2 * p3; p28 = 1 + 2^4 * p3; p27 = 1 + 2^2 * p1^3; p26 = 1 + 2 * p15; p25 = 1 + 2 * p1 * p6; p24 = 1 + 2^2 * p2^2; p23 = 1 + 2^5 * p1; p22 = 1 + 2 * p12; p21 = 1 + 2 * p1 * p5; p20 = 1 + 2^3 * p1^2; p19 = 1 + 2 * p2 * p3; p18 = 1 + 2 * p1 * p4; p17 = 1 + 2^2 * p1 * p2; p16 = 1 + 2 * p9; p15 = 1 + 2^2 * p5; p14 = 1 + 2 * p8; p13 = 1 + 2 * p1 * p3; p12 = 1 + 2^3 * p2; p11 = 1 + 2^2 * p1^2; p10 = 1 + 2 * p1 * p2; p9 = 1 + 2^2 * p3; p8 = 1 + 2 * p4; p7 = 1 + 2 * p1^2; p6 = 1 + 2^4; p5 = 1 + 2^2 * p1; p4 = 1 + 2 * p2; p3 = 1 + 2 * p1; p2 = 1 + 2^2; p1 = 1 + 2.
2^2203 - 1 = p264; p264 = 1 + 2 * p1^2 * p3 * p90 * p127 * p132 * p136 * p152 * p187 * p221 * p222 * p232 * p235 * p244 * p246 * p251 * p253 * p256 * p258 * p261 * p263; p263 = 1 + 2 * p1 * p18 * p39 * p51 * p67 * p81 * p262; p262 = 1 + 2^5 * p1 * p9 * p15 * p58 * p76 * p146 * p171 * p248 * p259; p261 = 1 + 2^3 * p1^3 * p51 * p72 * p93 * p260; p260 = 1 + 2 * p1 * p3 * p10 * p15 * p27 * p41 * p234 * p254 * p257; p259 = 1 + 2 * p1 * p66 * p149 * p210 * p250 * p255; p258 = 1 + 2^6 * p1 * p2^3 * p51 * p96 * p162 * p241 * p243; p257 = 1 + 2^2 * p3^3 * p54 * p216 * p230 * p242; p256 = 1 + 2^3 * p2 * p6 * p51 * p225 * p252; p255 = 1 + 2^5 * p3^2 * p4 * p25 * p185 * p211 * p236; p254 = 1 + 2^2 * p3 * p32 * p223 * p239; p253 = 1 + 2 * p1 * p11 * p51 * p247; p252 = 1 + 2 * p3 * p14 * p249; p251 = 1 + 2 * p2 * p3 * p7 * p51 * p112 * p237; p250 = 1 + 2 * p1 * p2 * p219 * p231; p249 = 1 + 2^3 * p1 * p2 * p245; p248 = 1 + 2 * p4 * p14 * p150 * p164 * p197 * p206; p247 = 1 + 2^2 * p4 * p192 * p208 * p213; p246 = 1 + 2 * p1 * p11 * p51 * p205 * p226; p245 = 1 + 2^2 * p3 * p143 * p238; p244 = 1 + 2^8 * p1^3 * p2^2 * p51 * p79 * p176 * p214; p243 = 1 + 2^2 * p1 * p6 * p27 * p49 * p87 * p198 * p209; p242 = 1 + 2^3 * p1 * p121 * p140 * p145 * p165 * p180; p241 = 1 + 2 * p1 * p2 * p240; p240 = 1 + 2 * p1 * p13 * p18 * p137 * p228; p239 = 1 + 2 * p1^4 * p2 * p80 * p200 * p202; p238 = 1 + 2 * p1 * p2 * p30 * p161 * p167 * p204; p237 = 1 + 2^4 * p1^2 * p3 * p110 * p229; p236 = 1 + 2 * p55 * p233; p235 = 1 + 2 * p1 * p17 * p20 * p51 * p218; p234 = 1 + 2^2 * p2^3 * p116 * p220; p233 = 1 + 2^2 * p2 * p19 * p97 * p217; p232 = 1 + 2 * p1 * p51 * p105 * p215; p231 = 1 + 2^6 * p224; p230 = 1 + 2^2 * p1 * p103 * p122 * p188; p229 = 1 + 2^3 * p7 * p133 * p207; p228 = 1 + 2 * p227; p227 = 1 + 2^4 * p8 * p10 * p108 * p189; p226 = 1 + 2 * p77 * p168 * p177; p225 = 1 + 2 * p1 * p4 * p5 * p172 * p174; p224 = 1 + 2^3 * p1^3 * p4 * p19 * p46 * p183; p223 = 1 + 2 * p1^2 * p15 * p212; p222 = 1 + 2^3 * p5 * p20 * p50 * p51 * p159; p221 = 1 + 2^3 * p1 * p7 * p51 * p125 * p142; p220 = 1 + 2^3 * p1^2 * p70 * p201; p219 = 1 + 2^4 * p1 * p2 * p22 * p113 * p169; p218 = 1 + 2 * p1 * p18 * p163 * p166; p217 = 1 + 2 * p1 * p156 * p184; p216 = 1 + 2 * p1^2 * p155 * p179; p215 = 1 + 2^2 * p5 * p11 * p199; p214 = 1 + 2^3 * p1 * p3 * p22 * p34 * p52 * p94; p213 = 1 + 2^3 * p1 * p3 * p6 * p22 * p53 * p92; p212 = 1 + 2^2 * p14 * p27 * p62 * p129; p211 = 1 + 2 * p1^4 * p3 * p5^2 * p7 * p29 * p36; p210 = 1 + 2^2 * p1 * p139 * p153; p209 = 1 + 2 * p1^3 * p193; p208 = 1 + 2^2 * p7 * p15 * p29 * p134; p207 = 1 + 2 * p1 * p2 * p196; p206 = 1 + 2 * p6 * p194; p205 = 1 + 2^3 * p17 * p81 * p117; p204 = 1 + 2 * p203; p203 = 1 + 2^2 * p2 * p191; p202 = 1 + 2^2 * p1 * p78 * p157; p201 = 1 + 2 * p1^2 * p190; p200 = 1 + 2^2 * p1^3 * p181; p199 = 1 + 2 * p1^3 * p5 * p7 * p154; p198 = 1 + 2^2 * p195; p197 = 1 + 2^2 * p1 * p4 * p15 * p148; p196 = 1 + 2 * p120 * p131; p195 = 1 + 2^2 * p1^2 * p89 * p101; p194 = 1 + 2^3 * p1 * p7 * p11 * p126; p193 = 1 + 2 * p1 * p5 * p41 * p123; p192 = 1 + 2 * p1 * p6 * p16 * p138; p191 = 1 + 2 * p2 * p72 * p124; p190 = 1 + 2 * p186; p189 = 1 + 2^2 * p1^2 * p3^2 * p16 * p65; p188 = 1 + 2^2 * p1 * p2 * p3 * p158; p187 = 1 + 2^6 * p1^2 * p2 * p3^2 * p51; p186 = 1 + 2 * p182; p185 = 1 + 2 * p1^2 * p173; p184 = 1 + 2 * p3^3 * p141; p183 = 1 + 2 * p1 * p178; p182 = 1 + 2 * p2 * p175; p181 = 1 + 2 * p2 * p6 * p147; p180 = 1 + 2 * p1 * p57 * p104; p179 = 1 + 2 * p8 * p13 * p95; p178 = 1 + 2 * p2 * p4^2 * p96; p177 = 1 + 2^3 * p1^2 * p144; p176 = 1 + 2^2 * p170; p175 = 1 + 2^5 * p1 * p9 * p71; p174 = 1 + 2 * p1^2 * p2 * p9 * p68; p173 = 1 + 2 * p1^2 * p151; p172 = 1 + 2^2 * p1^2 * p135; p171 = 1 + 2^2 * p20 * p91; p170 = 1 + 2^2 * p1 * p31 * p45; p169 = 1 + 2^3 * p3 * p118; p168 = 1 + 2 * p160; p167 = 1 + 2^3 * p2^2 * p86; p166 = 1 + 2^4 * p1 * p114; p165 = 1 + 2^4 * p130; p164 = 1 + 2^3 * p1 * p5 * p72; p163 = 1 + 2 * p30 * p73; p162 = 1 + 2^2 * p2 * p3 * p82; p161 = 1 + 2 * p37 * p60; p160 = 1 + 2 * p8 * p102; p159 = 1 + 2 * p1 * p2 * p107; p158 = 1 + 2^2 * p1 * p8 * p61; p157 = 1 + 2^2 * p1^2 * p99; p156 = 1 + 2^2 * p1^2 * p98; p155 = 1 + 2^5 * p2 * p4 * p15; p154 = 1 + 2 * p1 * p128; p153 = 1 + 2 * p14 * p74; p152 = 1 + 2^3 * p1^3 * p51; p151 = 1 + 2^2 * p1 * p8 * p44; p150 = 1 + 2^2 * p1 * p3 * p7 * p14; p149 = 1 + 2 * p1 * p3 * p11 * p14; p148 = 1 + 2^3 * p1 * p2 * p63; p147 = 1 + 2 * p2 * p17 * p28; p146 = 1 + 2 * p1^3 * p4 * p27; p145 = 1 + 2^3 * p1^2 * p4 * p17; p144 = 1 + 2^2 * p3 * p4 * p33; p143 = 1 + 2 * p1 * p3 * p8 * p12; p142 = 1 + 2 * p1^2 * p2 * p56; p141 = 1 + 2 * p25 * p38; p140 = 1 + 2^2 * p119; p139 = 1 + 2^2 * p11 * p42; p138 = 1 + 2 * p1 * p111; p137 = 1 + 2^3 * p100; p136 = 1 + 2 * p1 * p5 * p51; p135 = 1 + 2^4 * p1^3 * p2 * p4; p134 = 1 + 2^2 * p1 * p88; p133 = 1 + 2^2 * p2^2 * p40; p132 = 1 + 2 * p1^3 * p51; p131 = 1 + 2^2 * p1 * p84; p130 = 1 + 2^2 * p1 * p83; p129 = 1 + 2^2 * p1 * p2^2 * p15; p128 = 1 + 2 * p115; p127 = 1 + 2 * p6 * p51; p126 = 1 + 2 * p1 * p2 * p3 * p16; p125 = 1 + 2^2 * p2 * p64; p124 = 1 + 2 * p1 * p7 * p26; p123 = 1 + 2^2 * p9 * p25; p122 = 1 + 2^2 * p2 * p3^2 * p4; p121 = 1 + 2 * p109; p120 = 1 + 2 * p106; p119 = 1 + 2^5 * p1 * p3 * p5; p118 = 1 + 2 * p5 * p47; p117 = 1 + 2 * p1 * p3 * p38; p116 = 1 + 2^2 * p3^2 * p11; p115 = 1 + 2 * p2 * p3 * p26; p114 = 1 + 2^2 * p1^2 * p2 * p11; p113 = 1 + 2 * p1^2 * p48; p112 = 1 + 2 * p1 * p75; p111 = 1 + 2 * p2 * p5 * p13; p110 = 1 + 2 * p1^5 * p4; p109 = 1 + 2^2 * p10 * p13; p108 = 1 + 2^2 * p4 * p29; p107 = 1 + 2^2 * p1 * p3 * p16; p106 = 1 + 2^2 * p3 * p5^2; p105 = 1 + 2^2 * p1 * p2 * p19; p104 = 1 + 2 * p1 * p7 * p11; p103 = 1 + 2^3 * p1 * p2^2 * p3; p102 = 1 + 2^2 * p1^2 * p27; p101 = 1 + 2 * p1 * p3 * p23; p100 = 1 + 2 * p4 * p5^2; p99 = 1 + 2 * p1^3 * p2 * p5; p98 = 1 + 2 * p85; p97 = 1 + 2^3 * p1^2 * p12; p96 = 1 + 2 * p1 * p59; p95 = 1 + 2^2 * p1 * p43; p94 = 1 + 2^5 * p22; p93 = 1 + 2 * p1^2 * p32; p92 = 1 + 2 * p4 * p28; p91 = 1 + 2 * p1 * p2 * p3 * p4; p90 = 1 + 2 * p1 * p51; p89 = 1 + 2^3 * p1 * p23; p88 = 1 + 2 * p2 * p4 * p6; p87 = 1 + 2^3 * p1^2 * p8; p86 = 1 + 2 * p73; p85 = 1 + 2^6 * p2^2; p84 = 1 + 2 * p1^2 * p21; p83 = 1 + 2^2 * p1 * p2 * p8; p82 = 1 + 2 * p69; p81 = 1 + 2 * p6 * p11; p80 = 1 + 2^2 * p1 * p25; p79 = 1 + 2 * p3 * p22; p78 = 1 + 2^2 * p1 * p22; p77 = 1 + 2 * p1^2 * p2 * p4; p76 = 1 + 2 * p4 * p13; p75 = 1 + 2^3 * p1^2 * p5; p74 = 1 + 2 * p1^2 * p3^2; p73 = 1 + 2^3 * p25; p72 = 1 + 2^2 * p1 * p16; p71 = 1 + 2^5 * p1 * p3; p70 = 1 + 2 * p3 * p14; p69 = 1 + 2^2 * p35; p68 = 1 + 2 * p1 * p27; p67 = 1 + 2^3 * p3 * p4; p66 = 1 + 2^2 * p1^2 * p6; p65 = 1 + 2^3 * p1 * p2^2; p64 = 1 + 2 * p5 * p8; p63 = 1 + 2^4 * p11; p62 = 1 + 2^2 * p1^3 * p2; p61 = 1 + 2 * p1^2 * p9; p60 = 1 + 2 * p1 * p22; p59 = 1 + 2 * p43; p58 = 1 + 2 * p5 * p6; p57 = 1 + 2 * p1 * p20; p56 = 1 + 2^4 * p1^3; p55 = 1 + 2^3 * p1 * p6; p54 = 1 + 2^2 * p1^2 * p4; p53 = 1 + 2^2 * p24; p52 = 1 + 2 * p1^3 * p3; p51 = 1 + 2 * p1 * p17; p50 = 1 + 2^5 * p4; p49 = 1 + 2^2 * p1 * p9; p48 = 1 + 2^4 * p1 * p3; p47 = 1 + 2^2 * p21; p46 = 1 + 2^3 * p1 * p5; p45 = 1 + 2^2 * p20; p44 = 1 + 2^3 * p2 * p3; p43 = 1 + 2^3 * p9; p42 = 1 + 2^2 * p1 * p7; p41 = 1 + 2 * p1 * p11; p40 = 1 + 2 * p1 * p2 * p3; p39 = 1 + 2 * p1^2 * p4; p38 = 1 + 2^2 * p1^2 * p2; p37 = 1 + 2 * p23; p36 = 1 + 2^2 * p13; p35 = 1 + 2 * p1^4; p34 = 1 + 2 * p1 * p2^2; p33 = 1 + 2^2 * p11; p32 = 1 + 2 * p1 * p8; p31 = 1 + 2^3 * p6; p30 = 1 + 2 * p1^2 * p3; p29 = 1 + 2^4 * p3; p28 = 1 + 2^2 * p1^3; p27 = 1 + 2 * p15; p26 = 1 + 2 * p1 * p6; p25 = 1 + 2^2 * p2^2; p24 = 1 + 2^5 * p1; p23 = 1 + 2^3 * p4; p22 = 1 + 2 * p12; p21 = 1 + 2 * p1 * p5; p20 = 1 + 2^3 * p1^2; p19 = 1 + 2 * p2 * p3; p18 = 1 + 2 * p1 * p4; p17 = 1 + 2^2 * p1 * p2; p16 = 1 + 2 * p9; p15 = 1 + 2^2 * p5; p14 = 1 + 2 * p8; p13 = 1 + 2 * p1 * p3; p12 = 1 + 2^3 * p2; p11 = 1 + 2^2 * p1^2; p10 = 1 + 2 * p1 * p2; p9 = 1 + 2^2 * p3; p8 = 1 + 2 * p4; p7 = 1 + 2 * p1^2; p6 = 1 + 2^4; p5 = 1 + 2^2 * p1; p4 = 1 + 2 * p2; p3 = 1 + 2 * p1; p2 = 1 + 2^2; p1 = 1 + 2.
factorization of the 10th Fermat number. the large powers of 2 always present in Fermat number factorizations are less obvious in this notation compared to RPPFN.
2^1024 + 1 = p107 * p116 * p140 * p144; p144 = 1 + 2^13 * p1 * p5 * p8 * p9 * p64 * p112 * p137 * p141 * p142 * p143; p143 = 1 + 2 * p2 * p127 * p132 * p134 * p136; p142 = 1 + 2^2 * p3^2 * p5 * p56 * p126 * p139; p141 = 1 + 2^2 * p105 * p138; p140 = 1 + 2^12 * p1 * p63 * p66 * p96 * p130; p139 = 1 + 2^10 * p113 * p131; p138 = 1 + 2 * p5 * p95 * p121 * p125; p137 = 1 + 2^2 * p3 * p51 * p135; p136 = 1 + 2^2 * p1 * p69 * p97 * p129; p135 = 1 + 2^5 * p1 * p9 * p133; p134 = 1 + 2 * p46 * p118 * p122; p133 = 1 + 2^2 * p1^3 * p5 * p18 * p128; p132 = 1 + 2^3 * p1^2 * p52 * p109 * p119; p131 = 1 + 2 * p1 * p3 * p31 * p114 * p115; p130 = 1 + 2 * p59 * p75 * p82 * p117; p129 = 1 + 2^2 * p2 * p103 * p124; p128 = 1 + 2 * p1^3 * p3 * p45 * p58 * p92 * p94; p127 = 1 + 2^2 * p2 * p3^2 * p5 * p123; p126 = 1 + 2 * p10 * p102 * p108; p125 = 1 + 2^5 * p1 * p2 * p10 * p26 * p70 * p84; p124 = 1 + 2^3 * p1 * p2 * p7 * p42 * p61 * p67; p123 = 1 + 2 * p2 * p13 * p120; p122 = 1 + 2^4 * p2 * p5 * p14 * p27 * p99; p121 = 1 + 2^3 * p2 * p11 * p19 * p98; p120 = 1 + 2^5 * p2 * p110; p119 = 1 + 2^3 * p1 * p3 * p53 * p89; p118 = 1 + 2^2 * p1 * p5 * p7 * p104; p117 = 1 + 2^5 * p1^2 * p4 * p11 * p88; p116 = 1 + 2^14 * p1^2 * p9 * p11 * p12; p115 = 1 + 2 * p1 * p111; p114 = 1 + 2^3 * p1 * p3 * p48 * p74; p113 = 1 + 2^3 * p1^2 * p20 * p34 * p40; p112 = 1 + 2^2 * p1^3 * p44 * p68; p111 = 1 + 2 * p55 * p91; p110 = 1 + 2^3 * p1 * p4 * p101; p109 = 1 + 2 * p3 * p14 * p23 * p49; p108 = 1 + 2 * p106; p107 = 1 + 2^12 * p71; p106 = 1 + 2 * p24 * p93; p105 = 1 + 2^2 * p2 * p3 * p87; p104 = 1 + 2 * p1 * p100; p103 = 1 + 2^2 * p11 * p72; p102 = 1 + 2 * p1^2 * p2^2 * p14^2; p101 = 1 + 2^2 * p1 * p6 * p62; p100 = 1 + 2 * p1^2 * p2^2 * p54; p99 = 1 + 2 * p1 * p5 * p73; p98 = 1 + 2 * p2^2 * p6 * p9^2; p97 = 1 + 2 * p1 * p90; p96 = 1 + 2 * p1^2 * p78; p95 = 1 + 2 * p1 * p81; p94 = 1 + 2^3 * p1 * p65; p93 = 1 + 2 * p2 * p9 * p37; p92 = 1 + 2 * p86; p91 = 1 + 2 * p85; p90 = 1 + 2^5 * p1 * p2 * p30; p89 = 1 + 2 * p1^5 * p28; p88 = 1 + 2 * p83; p87 = 1 + 2^2 * p8 * p38; p86 = 1 + 2 * p80; p85 = 1 + 2 * p79; p84 = 1 + 2 * p77; p83 = 1 + 2 * p2 * p60; p82 = 1 + 2^2 * p1 * p2 * p3 * p17; p81 = 1 + 2 * p76; p80 = 1 + 2^2 * p8 * p32; p79 = 1 + 2^3 * p2 * p39; p78 = 1 + 2 * p1 * p2 * p4 * p17; p77 = 1 + 2 * p5 * p41; p76 = 1 + 2^7 * p3 * p6; p75 = 1 + 2^5 * p2 * p18; p74 = 1 + 2^2 * p1 * p2 * p4 * p7; p73 = 1 + 2 * p1 * p2 * p35; p72 = 1 + 2 * p1^3 * p4 * p7; p71 = 1 + 2 * p1 * p2 * p3 * p13; p70 = 1 + 2 * p1 * p2^2 * p16; p69 = 1 + 2 * p1^3 * p5^2; p68 = 1 + 2^6 * p3 * p7; p67 = 1 + 2 * p1 * p6 * p19; p66 = 1 + 2 * p2 * p43; p65 = 1 + 2 * p1 * p47; p64 = 1 + 2^3 * p3 * p25; p63 = 1 + 2 * p57; p62 = 1 + 2^2 * p1 * p35; p61 = 1 + 2^2 * p1 * p2 * p16; p60 = 1 + 2^2 * p1 * p33; p59 = 1 + 2 * p1^3 * p14; p58 = 1 + 2 * p5 * p25; p57 = 1 + 2 * p50; p56 = 1 + 2 * p1 * p2 * p18; p55 = 1 + 2^2 * p1^4 * p3; p54 = 1 + 2^2 * p1^3 * p7; p53 = 1 + 2^2 * p36; p52 = 1 + 2^4 * p20; p51 = 1 + 2 * p1^6; p50 = 1 + 2^7 * p4; p49 = 1 + 2 * p40; p48 = 1 + 2^2 * p1 * p22; p47 = 1 + 2^4 * p1 * p2^2; p46 = 1 + 2^2 * p1 * p2 * p6; p45 = 1 + 2 * p1^2 * p3^2; p44 = 1 + 2 * p3 * p14; p43 = 1 + 2 * p1 * p26; p42 = 1 + 2^2 * p2^2 * p3; p41 = 1 + 2 * p6 * p7; p40 = 1 + 2^7 * p2; p39 = 1 + 2 * p1 * p21; p38 = 1 + 2 * p5 * p8; p37 = 1 + 2^2 * p2 * p8; p36 = 1 + 2^3 * p1 * p6; p35 = 1 + 2 * p29; p34 = 1 + 2^2 * p18; p33 = 1 + 2 * p1^3 * p2; p32 = 1 + 2^2 * p15; p31 = 1 + 2 * p1 * p2 * p3; p30 = 1 + 2^2 * p3^2; p29 = 1 + 2 * p2 * p7; p28 = 1 + 2 * p1 * p2^2; p27 = 1 + 2 * p1 * p8; p26 = 1 + 2^3 * p6; p25 = 1 + 2^4 * p3; p24 = 1 + 2^2 * p1^3; p23 = 1 + 2 * p13; p22 = 1 + 2 * p1 * p6; p21 = 1 + 2^2 * p2^2; p20 = 1 + 2^5 * p1; p19 = 1 + 2 * p12; p18 = 1 + 2 * p1 * p5; p17 = 1 + 2^3 * p1^2; p16 = 1 + 2 * p2 * p3; p15 = 1 + 2 * p1 * p4; p14 = 1 + 2 * p9; p13 = 1 + 2^2 * p5; p12 = 1 + 2^3 * p2; p11 = 1 + 2^2 * p1^2; p10 = 1 + 2 * p1 * p2; p9 = 1 + 2^2 * p3; p8 = 1 + 2 * p4; p7 = 1 + 2 * p1^2; p6 = 1 + 2^4; p5 = 1 + 2^2 * p1; p4 = 1 + 2 * p2; p3 = 1 + 2 * p1; p2 = 1 + 2^2; p1 = 1 + 2.
10th number of the Sylvester's sequence.
sylvester[10] = 27392450308603031423410234291674686281194364367580914627947367941608692026226993634332118404582438634929548737283992369758487974306317730580753883429460344956410077034761330476016739454649828385541500213920807 = p53 * p78 * p111 * p115; p115 = 1 + 2^2 * p1 * p9 * p71 * p102 * p110 * p114; p114 = 1 + 2 * p1^3 * p4 * p97 * p103 * p113; p113 = 1 + 2 * p1 * p65 * p83 * p112; p112 = 1 + 2 * p1^2 * p3 * p38 * p93 * p101 * p107; p111 = 1 + 2 * p1 * p58 * p82 * p95 * p108; p110 = 1 + 2 * p3 * p10 * p67 * p109; p109 = 1 + 2 * p1 * p64 * p106; p108 = 1 + 2 * p1 * p15 * p45 * p89 * p90; p107 = 1 + 2^5 * p23 * p105; p106 = 1 + 2 * p1 * p12 * p104; p105 = 1 + 2^7 * p2 * p12 * p26 * p61 * p88; p104 = 1 + 2 * p2 * p16 * p100; p103 = 1 + 2 * p1 * p43 * p99; p102 = 1 + 2 * p2 * p4 * p17 * p29 * p63 * p85; p101 = 1 + 2^6 * p24 * p98; p100 = 1 + 2 * p2 * p18 * p52 * p92; p99 = 1 + 2 * p1 * p3 * p28 * p69 * p74; p98 = 1 + 2^4 * p1^2 * p73 * p77; p97 = 1 + 2 * p96; p96 = 1 + 2^2 * p94; p95 = 1 + 2^2 * p3 * p34 * p81; p94 = 1 + 2 * p1 * p3 * p91; p93 = 1 + 2 * p2 * p3 * p4 * p84; p92 = 1 + 2^4 * p1 * p2^2 * p4 * p68; p91 = 1 + 2 * p2 * p87; p90 = 1 + 2^4 * p86; p89 = 1 + 2^3 * p1^2 * p3 * p42 * p48; p88 = 1 + 2 * p1^2 * p80; p87 = 1 + 2^7 * p1 * p70; p86 = 1 + 2 * p1 * p2 * p5 * p30 * p40; p85 = 1 + 2 * p27 * p66; p84 = 1 + 2^2 * p79; p83 = 1 + 2 * p4 * p17 * p57; p82 = 1 + 2 * p49 * p51; p81 = 1 + 2 * p3 * p6 * p62; p80 = 1 + 2^2 * p1 * p31 * p50; p79 = 1 + 2 * p1 * p76; p78 = 1 + 2 * p1 * p75; p77 = 1 + 2^2 * p9 * p18 * p19; p76 = 1 + 2^4 * p26 * p33; p75 = 1 + 2 * p1 * p2 * p60; p74 = 1 + 2 * p72; p73 = 1 + 2^2 * p8 * p55; p72 = 1 + 2 * p35 * p39; p71 = 1 + 2 * p8 * p54; p70 = 1 + 2^2 * p11 * p47; p69 = 1 + 2 * p2 * p3^2 * p4 * p6; p68 = 1 + 2^2 * p1 * p2 * p3 * p32; p67 = 1 + 2^16; p66 = 1 + 2 * p1 * p59; p65 = 1 + 2^2 * p2 * p8 * p22; p64 = 1 + 2 * p1^3 * p44; p63 = 1 + 2^8 * p3 * p5; p62 = 1 + 2^5 * p1^2 * p14; p61 = 1 + 2^2 * p56; p60 = 1 + 2 * p1^2 * p46; p59 = 1 + 2^2 * p1^4 * p9; p58 = 1 + 2^2 * p1 * p41; p57 = 1 + 2^3 * p1^2 * p3^2; p56 = 1 + 2^2 * p1 * p36; p55 = 1 + 2^2 * p2 * p4 * p5; p54 = 1 + 2^3 * p1 * p25; p53 = 1 + 2 * p1^2 * p26; p52 = 1 + 2 * p1^3 * p11; p51 = 1 + 2^4 * p22; p50 = 1 + 2 * p1^2 * p19; p49 = 1 + 2^7 * p1^2; p48 = 1 + 2^2 * p1^2 * p8; p47 = 1 + 2^2 * p1 * p16; p46 = 1 + 2^2 * p2^2 * p3; p45 = 1 + 2^2 * p1^2 * p6; p44 = 1 + 2 * p37; p43 = 1 + 2 * p1 * p21; p42 = 1 + 2 * p2 * p13; p41 = 1 + 2 * p1^3 * p3; p40 = 1 + 2^2 * p1 * p10; p39 = 1 + 2^4 * p1 * p3; p38 = 1 + 2^2 * p20; p37 = 1 + 2^3 * p2 * p3; p36 = 1 + 2^2 * p18; p35 = 1 + 2^8; p34 = 1 + 2^2 * p3^2; p33 = 1 + 2^6 * p1; p32 = 1 + 2 * p2 * p7; p31 = 1 + 2 * p21; p30 = 1 + 2 * p1 * p2^2; p29 = 1 + 2^2 * p11; p28 = 1 + 2^3 * p6; p27 = 1 + 2 * p2 * p5; p26 = 1 + 2 * p1^2 * p3; p25 = 1 + 2^4 * p3; p24 = 1 + 2^2 * p1^3; p23 = 1 + 2 * p1 * p6; p22 = 1 + 2^5 * p1; p21 = 1 + 2 * p12; p20 = 1 + 2 * p1 * p5; p19 = 1 + 2 * p2 * p3; p18 = 1 + 2 * p1 * p4; p17 = 1 + 2^2 * p1 * p2; p16 = 1 + 2 * p9; p15 = 1 + 2^2 * p5; p14 = 1 + 2 * p8; p13 = 1 + 2 * p1 * p3; p12 = 1 + 2^3 * p2; p11 = 1 + 2^2 * p1^2; p10 = 1 + 2 * p1 * p2; p9 = 1 + 2^2 * p3; p8 = 1 + 2 * p4; p7 = 1 + 2 * p1^2; p6 = 1 + 2^4; p5 = 1 + 2^2 * p1; p4 = 1 + 2 * p2; p3 = 1 + 2 * p1; p2 = 1 + 2^2; p1 = 1 + 2.
Cunningham chain of the first kind, length 17.
CC17 is still the longest known chain of the first kind, according to http://primerecords.dk/Cunningham_Chain_records.htm
p33 = 180868411173855991792926719 = 1 + 2 * p32
p32 = 90434205586927995896463359 = 1 + 2 * p31
p31 = 45217102793463997948231679 = 1 + 2 * p30
p30 = 22608551396731998974115839 = 1 + 2 * p29
p29 = 11304275698365999487057919 = 1 + 2 * p28
p28 = 5652137849182999743528959 = 1 + 2 * p27
p27 = 2826068924591499871764479 = 1 + 2 * p26
p26 = 1413034462295749935882239 = 1 + 2 * p25
p25 = 706517231147874967941119 = 1 + 2 * p24
p24 = 353258615573937483970559 = 1 + 2 * p23
p23 = 176629307786968741985279 = 1 + 2 * p22
p22 = 88314653893484370992639 = 1 + 2 * p21
p21 = 44157326946742185496319 = 1 + 2 * p20
p20 = 22078663473371092748159 = 1 + 2 * p19
p19 = 11039331736685546374079 = 1 + 2 * p18
p18 = 5519665868342773187039 = 1 + 2 * p17
p17 = 2759832934171386593519 = 1 + 2 * p8 * p14 * p15 * p16
p16 = 27932129 = 1 + 2^5 * p3 * p6 * p12
p15 = 1915591 = 1 + 2 * p1 * p2 * p13
p14 = 265873 = 1 + 2^4 * p1 * p7 * p9
p13 = 63853 = 1 + 2^2 * p1 * p5 * p11
p12 = 6563 = 1 + 2 * p5 * p10
p11 = 313 = 1 + 2^3 * p1 * p4
p10 = 193 = 1 + 2^6 * p1
p9 = 191 = 1 + 2 * p2 * p6
p8 = 97 = 1 + 2^5 * p1
p7 = 29 = 1 + 2^2 * p3
p6 = 19 = 1 + 2 * p1^2
p5 = 17 = 1 + 2^4
p4 = 13 = 1 + 2^2 * p1
p3 = 7 = 1 + 2 * p1
p2 = 5 = 1 + 2^2
p1 = 3 = 1 + 2
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