Talent s error. of Geometry". He was active in Alexandria during the reign. of Ptolemy I ( BC). His Elements is one of the most influential

Transcription

1 Talent s error 1. Introduce Euclid was a Gree mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I ( BC). His Elements is one of the most influential wors in the history of mathematics, serving as the main textboo for teaching mathematics(esecially geometry) from the time of its ublication until the late 19th or early 20th century. In the Elements, Euclid deduced the rinciles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote wors on ersective, conic sections, sherical geometry, number theory and rigor. His methodology has also influenced many great scientists. Most conclusions of a mathematical talent are correct. Peole adore him. But if a talent has made an imercetible mistae, most eole will trust it was correct also. 2. Prime is infinite One of Euclid s famous roof is rime is infinite. Suose rime is finite, P = { 2, 3, 5 }. Constructing a number + 1 = All of 2, 3, 5,, can t divide

2 + 1. Either 1 + is a bigger rime or + 1 is a comosite number that can resolve a rime being bigger than. So suosition is false, rime is infinite. It s a clever roof. 3. Prime is finite The interesting thing is lie below. Suose rime is infinite. Constructing a number = /( -1), is a rime, is not an integer, otherwise ( -1) can divide, then has found a divisor and is a comosite number. Because rime is infinite, = lim / ( 1) 1 - =, is an integer. ( -1) can divide found a divisor and is a comosite number. So suosition is false, rime is finite., then has 4. Who is wrong Both rime being finite and rime being infinite are correct? One of they must be wrong. But who is correct? Most eole will say rime being infinite is correct. But why, I thin the only reason is that it s Euclid s roof, no other more strong reason. Firstly, let me mae an exeriment. The density of rime ((count of rime)/(count of odd number)) is oscillating to trend to 0 when odd number is increasing.

3 Prime density data table lie below, which dislays the regularity very clearly. 2K+1 sum(rime)/k sum( ) / K sum( ) / K

4 The function lot lie below. finite. From the exeriment, it seems the evidence suorts rime being

5 5. Talent s error To find out the root cause, I have checed both roofs carefully. Please read the roof again. Constructing a number + 1 = All of 2, 3, 5,, can t divide 1 +. Either + 1 is a bigger rime or 1 + is a comosite number that can resolve a rime being bigger than. The bold section has included 3 rerequisites. 1. A number is either a rime, or a comosite number. 2. Nature number can never reach infinite. 3. A comosite number can always resolve a rime. For rerequisite 1, it s easy to find an excetion. Neither 1 is rime, nor comosite number. Is infinite a comosite number? Many eole don t thin infinite is a number. Then what s infinite? For rerequisite 2, if nature number can never reach infinite, does the distance from infinite become more and more big or small? If the distance from infinite is constant, it means that C = -n = -(n+1) Þ = -1 Þ = ( -1) -1 = -2, Þ = -n = C Þ C =, If the distance from infinite becomes more and more big, it means that -n < -(n+1) Þ +1 <, Þ 1 = lim( n+ 1) / n =( +1)/ <1Þ1<1. n

6 If the distance from infinite becomes more and more small, when ( ) < e, e is smaller than any number, it means it can be smaller than 1. If e <1, then ( >, Þ 1 + has exceeded infinite. ) < e <1, Þ + 1 = For rerequisite 3, When + 1 >, neither it is a rime, nor can resolve a bigger rime. So Euclid s roof is wrong. 6. Conclusion A great mathematician can hardly mae a mistae, but if it s an imercetible mistae, which will mislead mathematician for a long time because of eole s adoration. Though Euclid is a great mathematician, he is a human being also. Both the exeriment and logic analyze have demonstrated the famous roof is wrong. Whatever, Euclid is a mathematician with my full resects for his great achievements.

7 References [1] John Friedlander and Henry Iwaniec, The olynomial X2+ Y4 catures its rimes, 148 (1998), [2] John Friedlander and Henry Iwaniec, Asymtotic sieve for rimes, 148 (1998), [3] University of TongJi,Higher mathematics, 465(1991) [4] E. Bombieri, The asymtotic sieve, Mem. Acad. Naz. dei XL, 1/2 (1976), [5] W. Due, J.B. Friedlander, and H. Iwaniec, Equidistribution of roots of a quadratic congruence to rime moduli, Ann. of Math. 141 (1995), [6] C.L. Stewart and J. To, On rans of twists of ellitic curves and ower-free values of binary forms, J. Amer. Math. Soc. 8 (1995), [7] E. Fouvry and H. Iwaniec, Gaussian rimes, Acta Arith. 79 (1997), [8] J. Friedlander and H. Iwaniec, Bombieri's sieve, in Analytic Number Theory, Proc. Halberstam Conf., Allerton Par Illnois, June 1995, ed. B. C. Berndt et al., , BirhÄauser (Boston), [9], The olynomial X2 + Y 4 catures its rimes, Ann. of Math. 148 (1998), [10] G. Harman, On the distribution of modulo one, J. London Math. Soc. 27 (1983),9-18.