caption: Prove that there are an infinite number of prime numbers
exif: {"Image DocumentName": "Prove that there are an infinite number of prime numbers", "Image ImageDescription": "\ud835\udc6c\ud835\udc7c\ud835\udc6a\ud835\udc73\ud835\udc70\ud835\udc6b\ud835\udc94 \ud835\udc91\ud835\udc93\ud835\udc90\ud835\udc90\ud835\udc87. Prove that there are an infinite number of prime numbers. WB 10. Consider any finite list of prime numbers p1, p2, ..., pn. Let P be the product of all the prime numbers in the list: \ud835\udc43 = \ud835\udc5d 1 \u00d7 \ud835\udc5d 2 \u00d7 \ud835\udc5d 3 \u00d7\u2026 \u2026\u00d7 \ud835\udc5d \ud835\udc5b. P is not prime since it has n divisors (divisible by every integer up to pn ) Let Q = P+1 \ud835\udc44= \ud835\udc5d 1 \u00d7 \ud835\udc5d 2 \u00d7 \ud835\udc5d 3 \u00d7\u2026 \u2026\u00d7 \ud835\udc5d \ud835\udc5b +1. If Q is prime, then there is at least one more prime than is in the list. If Q is not prime, then some prime factor \ud835\udc5d \ud835\udc56 divides Q. If this factor \ud835\udc5d \ud835\udc56 were on our list, then it would divide P (since P is the product of every number on the list); but Q divides P + 1 ( = itself ) If \ud835\udc5d \ud835\udc56 divides P and Q, then \ud835\udc5d \ud835\udc56 would have to divide the difference of the two numbers, which is (P + 1) \u2212 P or just 1. Since no prime number divides 1, \ud835\udc5d \ud835\udc56 cannot be on the list. This means that at least one more prime number exists beyond those in the list. QED! \u2018I\u2019ve proved it !\u2019", "Image XResolution": "200", "Image YResolution": "200", "Image ResolutionUnit": "Pixels/Inch", "Image Software": "SlidePlayer", "Image YCbCrPositioning": "Centered"}