x^n+y^n=z^n当n>2时没有正整数解

小草

x^n+y^n=z^n当n>2时没有正整数解
         
                文/施承忠


证：
当n=1时
若x,y为任意正整数时，z就不能是任意的；当z和y为任意的正整数时，x就不能是任意的；当z和x为任意正整数时，y就不可能是任意的，所以x、y、z,至少有一个是不能任意的，它必须受其它两个正整数的约束。
当n=2时
x^2+y^2=z^2
我们只要证明y是任意素数时上式有整正数解就足够了.因为当y是合数时
设y=pq
若z^2-x^2=q^2成立,
则p^2z^2-p^2x^2=p^2q^2成立。
我们有
z^2-x^2=y^2
则z^2-x^2=(z-x)(z+x)
如果y是素数只有一种分解
y^2=1*y^2
则z-x=1
z+x=y^2
(y^2)-1/2=x
(y^2)+1/2=z
当n=3时
令y<x<z
(z^2-x^2)*y=y^3
=(z^2)*y-(x^2)*y=y^3
则
(z^2)*y<z^3
(x^2)*y<x^3
因为在z^2-x^2=y^2中
x=z-1
现在
3√((x^2)*y)=d<x
3√((z^2)*y)=e<z
z-2<d<z-1
x<e<x+1
因为x是正整数，x+1是正整数，e肯定不是正整数。
同样x也肯定不是正整数。
如果
(z^2)*y+m=z^3
(x^2)*y+m=f^3
则x<f<z,f肯定不是正整数。
如果
(x^2)*y+m=x^3
(z^2)*y+m=f^3
则x<f<z,结果相同。
所以x,z中至少有一个不是正整数。
令n=1时x=x1,n=n时x=xn
n=1时z=z1,n=n时z=zn
我们有
x-1<xn<xn-1<xn-2<...<x2<x
z-1<zn<zn-1<zn-2<...<z2<z
当n大于2时zn-xn<1
证毕。


其实当n很大时
(z*y^n-1)-(x*y^n-1)=y^n
则z→x→y,如果y是正整数，则z和x都不是正整数，经过变换z^n-xn^n=y^n,则xn→z，反之则
zn→x,所以z和x肯定其中有一个不是正整数。

y<x
则n√2y^n<z<n√2x^n

小草

7-4=3
z1=7
x1=4
y1=3
z1-x1=7-4=3

7*3-4*3=3^2
(21+4)-(12+4)=3^2
5^2-4^2=3^2
z2=5
x2=4
y2=3
z2-x2=1


25*3-16*3=3^3
z3=3√75=4.2172
x3=3√48=3.6342
y3=3
z3-x3=4.2172-3.6342=0.583

75+50=5^3
48+50=98
5^3-98=3^3
z3=3√125=5
x3=3√98=4.6104
y3=3
z3-x3=5-4.6104=0.6104

75+16=91
48+16=4^3
91-64=3^3
z3=3√91=4.4979
x3=3√64=4
y3=3
z3-x3=4.4979-4=0.4979

75*3-48*3=3^4
z4=4√225=3.873
x4=4√144=3.4641
y4=3
z4-x4=3.873-3.4641=0.4089

225+400=5^4
144+400=544
5^4-544=3^4
z4=4√625=5
x4=4√544=4.8295
y4=3
z4-x4=5-4.8295=0.1705

225+112=337
144+112=256
337-4^4=3^4
z4=4√337=4.2846
x4=4√256=4
y4=3
z4-x4=4.2846-4=0.2846

小草

7-3=4
z1=7
x1=3
y1=4
z1-x1=7-3=4

5^2-3^2=4^2
z2=5
x2=3
y2=4
z2-y2=5-4=1

25*4=100
9*4=36
100+25=5^3
36+25=61
5^3-61=4^3
z3=3√125=5
x3=3√61=3.9365
y3=3√64=4
z3-y3=5-4=1

小草

11-6=5
z1=11
x1=6
y1=5
z1-x1=11-6=4

z2=13
x2=12
y2=5
z2-x2=13-12=1

13^2*5=845
12^*5=720
845-720=5^3
z3=3√845=9.4541
x3=3√720=8.9628
y3=5
z3-x3=9.4541-8.9628=0.4913

845*5=4225
720*5=3600
4225-3600=5^4
z4=4√4225=8.0623
x4=4√3600=7.746
y4=5
z4-x4=8.0623-7.746=0.3163

4225*5=21125
3600*5=18000
21125-18000=5^5
z5=5√21125=7.3276
x5=5√18000=7.0967
y5=5
z5-x5=7.3276-7.0967=0.2309

21125*5=105625
18000*5=90000
105625-90000=5^6
z6=6√105625=6.8753
x6=6√90000=6.6943
y6=5
z6-x6=6.8753-=6.6943=0.191

105625*5=528125
90000*5=450000
528125-450000=5^7
z7=7√528125=6.5695
x7=7√450000=6.421
y7=5
z7-x7=6.5695-6.421=0.1486

528125*5=2640625
450000*5=2250000
2640625-2250000=5^8
z8=8√2640625=6.3491
x8=8√2250000=6.2233
y8=5
z8-x8=6.3491-6.2233=0.1258

2640625*5=13203125
2250000*5=11250000
13203125-11250000=5^9
z9=9√13203125=6.1828
x9=9√11250000=6.0738
y9=5
z9-x9=6.1828-6.0738=0.109

13203125*5=66015625
11250000*5=56250000
66015625-56250000=5^10
z10=10√66015625=6.0529
x10=10√56250000=5.9568
y10=5
z10-x10=6.0529-5.9568=0.0961

66015625*5=330078125
56250000*5=281250000
330078125-281250000=5^11
z11=11√330078125=5.9487
x1111√281250000=5.8627
y11=5
z11-x11=5.9487-5.8627=0.086

小草

15-8=7
z1=15
x1=8
y1=7
z1-x1=15-8=7

25^2-24^2=7^2
z2=25
x2=24
y2=7
z2-x2=25-24=1

25^2*7^14=423889420530625
24^2*7^14=390656489961024
423889420530625-390656489961024=7^16
z16=16√423889420530625=8.2074
x16=16√390656489961024=8.1656
y16=7
z16-x16=8.2074-8.1656=0.0418

小草

3^2=9
(5-4)(5+4)=9
z2=5
x2=4
y2=3
z2-x2=5-4=1
5^2*3-4^2*3=3^3
75-48=27
z3=3√75=4.2172
x3=3√48=3.6342
y3=3
z3-x3=4.2172-3.6342=0.583

4^2=16
(5-3)(5+3)=16
z2=5
x2=3
y2=4
z2-y2=5-4=1
5^2*4-3^2*4=4^3
100-36=64
z3=3√100=4.6416
x3=3√36=3.3019
y3=4
z3-y3=4.6416-4=0.6416

5^2=25
(13-12)(13+12)=25
z2=13
x2=12
y2=5
z2-x2=13-12=1
13^2*5-12^2*5=5^3
845-720=125
z3=3√845=9.4541
x3=3√720=8.9628
y3=5
z3-x3=9.4541-8.9628=0.4913

6^2=36
6=3*2
5^2*2^2-4^2*2^2=3^2*2^2
z2=5*2
x2=4*2
y2=3*2
z2/2-x2/2=5-4=1
5^2*3*2^3-4^2*3*2^3=3^3*2^3
75*8-48*8=216
z3=3√75*2=4.2172*2
x3=3√48*2=3.6342*2
y3=6
z3/2-x3/2=4.2172-3.6342=0.583

7^2=49
(25-24)(25+24)=49
z2=25
x2=24
y2=7
z2-x2=25-24=1
25^2*7-24^2*7=7^3
4375-4032=343
z3=3√4375=16.3553
x3=3√4032=15.9162
y3=7
z3-x3=16.3553-15.9162=0.4391

8^2=64
8=4*2
(5^2*2^2)-(3^2*2^2)=4^2*2^2
z2=5*2
x2=3*2
y2=4*2
z2/2-y2/2=5-4=1
5^2*4*2^3-3^2*4*2^3=8^3
800-288=512
z3=3√100*2=4.6416*2
x3=3√36*2=3.3019*2
y3=4*2
z3/2-y3/2=4.6416-4=0.6416


9^2=81
9=3*3
(5*2*3^2)-(4^2*3^2)=3^2*3^2
z2=5*3
x2=4*3
y2=3*3
z2/3-x2/3=5-4=1
5^2*3*3^3-4^2*3*3^3=3^3*3^3
2025-1296=729
z3=3√75*3=4.2172*3
x3=3√48*3=3.6342*3
y3=3*3
z3/3-x3/3=4.2172-3.6342=0.583


10^2=100
10=5*2
(13^2*2^2)-(12^2*2^2)=5^2*2^2
z2=13*2
x2=12*2
y2=5*2
z2/2-x2/2=13-12=1
13^2*5*2^3-12^2*5*2^3=5^3*2*3
6760-5760=1000
z3=3√845*2=9.4541*2
x3=3√720*2=8.9628*2
y3=5*2
z3/2-x3/2=9.4541-8.9628=0.4913

11^2=121
(61-60)(61+60)=121
z2=61
x2=60
y2=11
z2-x2=61-60=1
61^2*11-60^2*11=11^3
40931-39600=1331
z3=3√40931=34.4628
x3=3√39600=34.0851
y3=11
z3-x3=34.4628-34.0851=0.3777

12^2=144
12=3*4
(5^2*4^2)-(4^2*4^2)=3^2*4^2
z2=5*4
x2=4*4
y2=3*4
z2/4-x2/4=5-4=1
5^2*3*4^3-4^2*3*4^3=3^3*4^3
4800-3072=1728
z3=3√75*4=4.2172*4
x3=3√48*4=3.6342*4
y3=3*4
z3/4-x3/4=4.2172-3.6342=0.583

13^2=169
(85-84)(85+84)=169
z2=85
x2=84
y2=13
85^2*13-84^2*13=13^3
93925-91728=2197
z3=3√93925=45.4563
x3=3√91728=45.099
y3=13
z3-x3=45.4563-45.099=0.3573

14^2=196
14=7*2
(25^2*2^2)-(24^2*2*2)=7^2*2*2
z2=25*2
x2=24*2
y2=7*2
z2/2-x2/2=25-24=1
25^2*7*2^3-24^2*7*2^3=7^3*2^3
35000-32256=2744
z3=3√4375*2=16.3553
x3=3√4032*2=15.9162
y3=7*2
z3/2-x3/2=16.3553-15.9162=0.4391

15^2=225
3*5
(5^2*5^2)-(4^2*5^2)=3^2*5^2
z2=5*5
x2=4*5
y2=3*5
z2/5-x2/5=5-4=1
5^2*3*5^3-4^2*3*5^3=3^3*5^3
9375-6000=3375
z3=3√75*5=4.2172*5
x3=3√48*5=3.6342*5
y3=3*5
z3/5-x3/5=4.2172-3.6342=0.583

16^2=256
16=4*4
(5^2*4^2)-(3^2*4^2)=4^2*4^2
z2=5*4
x2=3*4
y2=4*4
z2/4-y2/4=5-4=1
5^2*4*4^3-3^2*4*4^3=4^3*4^3
6400-2304=4096
z3=3√100*4=4.6416*4
x3=3√36*4=3.3019*4
y3=4*4
z3-y3=4.6416-4=0.6416

17^2=289
(145-144)(145+144)=289
z2=145
x2=144
y2=17
z2-x2=145-144
145^2*17-144^2*17=17^3
357425-352512=4913
x3=3√357425=70.9678
x3=3√352512=70.6412
y3=17
z3-x3=70.9678-70.6412=0.3266

18^2=324
18=3*6
(5^2*6^2)-(4^2*6^2)=3^2*6^2
z2=5*6
x2=4*6
y2=3*6
z2-x2=z2/6-x2/6=5-4=1
5^2*3*6^3-4^2*3*6^3=3^3*6^3
16200-10368=5832
z3=3√75*6=4.2172*6
x3=3√48*6=3.6342*6
y3=3*6
z3/6-x3/6=4.2172-3.6342=0.583

19^2=361
(181-180)(181+180)=361
z2=181
x2=180
y2=19
z2-x2=181-180=1
181^2*19-180^2*19=19^3
622459-615600=6859
z3=3√622459=85.3828
x3=3√615600=85.068
y3=19
z3-x3=85.3828-85.068=0.3148

20^2=400
20=5*4
(13^2*4^2)-(12^2*4^2)=5^2*4^2
z2=13*4
x2=12*4
y2=5*4
z2/4-x2/4=13-12=1
13^2*5*4^3-12^2*5*4^3=5^3*4^3
54080-46080=8000
z3=3√845*4=9.4541
x3=3√720*4=8.9628
y3=5*4
z3/4-x3/4=9.4541-8.9628=0.4913

小草

21=3*7
5^2*7^2-4^2*7*2=3^2*7^2
1225-784=441
z2/7=5
x2/7=4
y2/7=3
z2/7-x2/7=5-4=1
5^2*3*7^3-4^2*3*7^3=3^3*7^3
25725-16464=9261
z3/7=3√75=4.2172
x3/7=3√48=3.6342
y3/7=3
z3/7-x3/7=4.2172-3.6342=0.583

小草

22=11*2
61^2*2^2-60^2*2^2=11^2*2^2
14884-14400=484
z2/2=61
x2/2=60
y2/2=11
z2/2-x2/2=61-60=1
61^2*11*2^3-60^2*11*2^3=11^3*2^3
327448-316800=10648
z3/2=3√40931=34.4628
x3/2=3√39600=34.0851
y3/2=11
z3/2-x3/2=34.4628-34.0851=0.3777

小草

23^2=529
(265-264)(265+264)=529
z2=265
x2=264
y2=23
z2-x2=265-264=1
265^2*23-264^2*23=23^3
1615175-1603008=12167
z3=3√1615175=117.3293
x3=3√1603008=117.034
y3=23
z2-x2=117.3293-117.034=0.2953

小草

24=3*8
5^2*8^2-4^2*8^2=3^2*8^2
1600-1024=576
z2/8=5
x2/8=4
y2/8=3
z2/8-x2/8=5-4=1
5^2*3*8^3-4^2*3*8^3=3^3*8^3
38400-24576=13824
z3/8=3√75=4.2172
x3/8=3√48=3.6342
y3/8=3
z3/8-x3/7=4.2172-3.6342=0.583