This proof asks one to assume there are a finite number of primes. If one then takes the product of all these numbers and adds one, that number must be prime, thereby proving the assumption false.
The product of prime number up to a certain one is called the primorial of that prime, after Harvey Dubner.
Primorials get very large, very quickly.
Here are the first 50 prime numbers and their primorials:
2: 2
3: 6
5: 30
7: 210
11: 2310
13: 30030
17: 510510
19: 9699690
23: 223092870
29: 6469693230
31: 200560490130
37: 7420738134810
41: 304250263527210
43: 13082761331670030
47: 614889782588491410
53: 32589158477190044730
59: 1922760350154212639070
61: 117288381359406970983270
67: 7858321551080267055879090
71: 557940830126698960967415390
73: 40729680599249024150621323470
79: 3217644767340672907899084554130
83: 267064515689275851355624017992790
89: 23768741896345550770650537601358310
97: 2305567963945518424753102147331756070
101: 232862364358497360900063316880507363070
103: 23984823528925228172706521638692258396210
107: 2566376117594999414479597815340071648394470
109: 279734996817854936178276161872067809674997230
113: 31610054640417607788145206291543662493274686990
127: 4014476939333036189094441199026045136645885247730
131: 525896479052627740771371797072411912900610967452630
137: 72047817630210000485677936198920432067383702541010310
139: 10014646650599190067509233131649940057366334653200433090
149: 1492182350939279320058875736615841068547583863326864530410
151: 225319534991831177328890236228992001350685163362356544091910
157: 35375166993717494840635767087951744212057570647889977422429870
163: 5766152219975951659023630035336134306565384015606066319856068810
167: 962947420735983927056946215901134429196419130606213075415963491270
173: 166589903787325219380851695350896256250980509594874862046961683989710
179: 29819592777931214269172453467810429868925511217482600306406141434158090
181: 5397346292805549782720214077673687806275517530364350655459511599582614290
191: 1030893141925860008499560888835674370998623848299590975192766715520279329390
193: 198962376391690981640415251545285153602734402721821058212203976095413910572270
197: 39195588149163123383161804554421175259738677336198748467804183290796540382737190
199: 7799922041683461553249199106329813876687996789903550945093032474868511536164700810
211: 1645783550795210387735581011435590727981167322669649249414629852197255934130751870910
223: 367009731827331916465034565550136732339800312955331782619462457039988073311157667212930
227: 83311209124804345037562846379881038241134671040860314654617977748077292641632790457335110
229: 19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190
External References