Ïîñëåäíÿÿ òåîðåìà Ôåðìà – ðåøåíèå â îáùåì âèäå
Ñåðãèí Ãåííàäèé Èâàíîâè÷,
âðà÷–ñòîìàòîëîã.
Òåîðåìà
Óðàâíåíèå xn+yn=zn ïðè n>2 íå èìååò ðåøåíèé â ðàöèîíàëüíûõ ÷èñëàõ, xyz≠0.
Âàðèàíò ¹1 (÷åðåç ïðîïîðöèþ).
Ïóñòü: x+y=z , x2+y2=z2 , xn-1+yn-1=zn-1, xn+yn=zn ,
y=z –x, y2=z2–x2, yn-1= zn-1–xn-1, yn=zn–xn.
Òîãäà:
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x2 /x=x, xn /xn-1=x; Ïðîïîðöèîíàëüíîå óðàâíåíèå ¹1 x2 /x=xn /xn-1 → x=xn /xn-1 → xxn-1=xn → xn=xn → x=x |
z2/z=z, zn/ zn-1=z Ïðîïîðöèîíàëüíîå óðàâíåíèå ¹2 z2/z = zn/zn-1 → z=zn /zn-1 → zzn-1=zn → zn=zn → z=z
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ïðîïîðöèîíàëüíîå óðàâíåíèå ¹3
Äîêàçàòåëüñòâî
(z2 –x2) /(z–x)=(zn –xn) /(zn-1 –xn-1) → (z+x)(z –x) /(z –x)=(zn –xn) /(zn-1 –xn-1) →
(z+x)=(zn –xn) /(zn-1 –xn-1) → (z+x)(zn-1 –xn-1)=zn –xn →
zn –zxn-1+zn-1x–xn = zn –xn → zn–zxn-1+zn-1x–xn–zn+xn=0 → –zxn-1+zn-1x=0 →
zn-1x=zxn-1 → zn-1x/ zx=zxn-1/ zx → zn-2=xn-2 → z=x → zn=xn →
zn–xn=0 → yn=zn–xn →
yn=0 →
y=0 →
xyz=0
ïðîòèâîðå÷èò óñëîâèþ
ïðîâåðî÷íûé âàðèàíò äëÿ n = 9
(z2–x2) /(z–x)=(z9 –x9) /(z8–x8) → (z+x)(z –x) /(z –x)=(z9–x9) /(z8–x8) →
(z+x)=(z9–x9) /(z8–x8) → (z+x)(z8–x8) = z9–x9 → z9–zx8+z8x–x9=z9–x9 →
z9–zx8+z8x–x9–z9+x9=0 → –zx8+z8x=0 → z8x=zx8 → z8x/zx = zx8/zx →
z7=x7 → z=x → z9=x9 → z9–x9=0 → y9= z9–x9 → y9=0 → y=0 →
xyz=0
ïðîòèâîðå÷èò óñëîâèþ
Âàðèàíò ¹2 (÷åðåç áèíîì Íüþòîíà).
Ïóñòü:
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xn+yn=zn x2+y2=z2 x+y=z |
yn= zn–xn y2= z2–x2 y= z–x |
xa0=x1 xa1=x2 xan-1=xn |
yb0=y1 yb1=y2 ybn-1=yn |
zc0=z1 zc1=z2 zcn-1=zn |
Òîãäà:
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a=x2 /x → a=x → xan-1=xn |
c=z2/z → c=z → zcn-1=zn |
b=y2 /y →
b=(z2 –x2) /(z –x) → b=(z+x)(z –x) /(z –x) → b=(z+x) →
y(z+x)n-1=yn → (z –x)(z+x)n-1=yn →
(z –x)(z+x) n-1=zn –xn
ïðè n=1
(z–x)(z+x)n-1=zn–xn → (z –x)(z+x)0=z –x → z–x=z –x
ïðè n=2
(z –x)(z+x)n-1=zn –xn → (z –x)(z+x) 1=z2–x2 → z2–x2=z2–x2
ïðè n=3 (äîêàçàòåëüñòâî)
(z –x)(z+x)n-1=zn–xn → (z –x)(z+x)2=z3–x3 →
(z –x)(z+x)2 = (z –x)( z2+zx+x2) → (z+x)2=(z2+zx+x2) →
z2+2zx+x2=z2+zx+x2 → zx=0
Åñëè y>0, òî z=y , x=0, xyz=0 → ïðîòèâîðå÷èò óñëîâèþ.
ïðè n=4 (äîêàçàòåëüñòâî)
(z–x)(z+x)n-1=zn–xn → (z–x)(z+x)3=z4–x4 →
(z –x)(z3+3z2x+3zx2+x3) =(z–x)(z3+z2x+zx2+x3) →
z3+3z2x+3zx2+x3=z3+z2x+zx2+x3 → 3z2x+3zx2=z2x+zx2 → 2z2x+2zx2=0 →
2zx(z+x)=0 → zx=0/2(z+x) → zx=0
Åñëè y>0, òî z=y, x=0, xyz=0 → ïðîòèâîðå÷èò óñëîâèþ.
ïðè n=5 (äîêàçàòåëüñòâî)
(z –x)(z+x)n-1=zn–xn → (z–x)(z+x)4=z5–x5 →
(z –x)(z4+4z3x+6 z2x2+4zx3+x4)=(z–x)(z4+z3x+z2x2+zx3+x4) →
z4+4z3x+6 z2x2+4zx3+ x4=z4+z3x+z2x2+zx3+x4 →
4z3x+6z2x2+4zx3 = z3x+z2x2+ zx3 → 3z3x+5 z2x2+3zx3=0 →
3zx(z2+2zx+x2)=0 → 3zx(z+x)2=0 → zx=0/3(z+x)2 → zx=0
Åñëè y>0, òî z=y, x=0, xyz=0 → ïðîòèâîðå÷èò óñëîâèþ.
ïðè n>2 (äîêàçàòåëüñòâî)
(z –x)(z+x)n-1=zn–xn
(n–2)zx((z+x)n-1– (zn –xn)/(z–x)) =0
zx=0/(n–2)((z+x)n-1–(zn –xn)/(z–x))
zx=0
Åñëè y>0, òî z=y, x=0.
xyz=0 → ïðîòèâîðå÷èò óñëîâèþ.
Òàê êàê ïîñëåäíÿÿ òåîðåìà Ôåðìà ÿâëÿåòñÿ ÷àñòíûì ñëó÷àåì èç, âàðèàíòîâ ¹1 è ¹2, â àëüòåðíàòèâó, êàê ñëåäñòâèå èç âûøåèçëîæåííîãî, ïðåäñòàâëÿþ ÷àñòíûé ñëó÷àé äëÿ òåîðåìû Ïèôàãîðà:
Óðàâíåíèå x2+y2=z2 ïðåäñòàâëåííîå â âèäå:
Ôîðìóëà¹1 (k(y2–1)/2)2+(ky)2=(k((y2–1)/2+1))2
ïðè k=íàòóðàëüíîìó ÷èñëó è ïðè y=íå÷åòíîìó íàòóðàëüíîìó ÷èñëó >1 ïðåäñòàâëÿåò ñîáîé âñå ñóùåñòâóþùèå ðåøåíèÿ èñêëþ÷èòåëüíî â íàòóðàëüíûõ ÷èñëàõ, xyz=íàòóðàëüíîìó ÷èñëó.
Ïðèìåð ¹1: k=8 y=13
(8*(132–1)/2)2+(8*13)2=(8*((132–1)/2+1))2 → 6722+1042=6802
Âîçíèêàåò ïîñëåäíèé âîïðîñ: Ãäå äîëæåí íàõîäèòüñÿ yn äëÿ ñîõðàíåíèÿ ñâîåé ñòåïåííîé çàâèñèìîñòè îò z è x? yn èìååò ñòðîãî êâàäðàòíóþ çàâèñèìîñòü îò z è x, è îòâåò äàåò óðàâíåíèå âèäà:
x2+yn=z2
Ïðèìåð ¹2: x=4 z=5
42+91 =52 122+92=152 362+93=452 1082+94=1352
(4*3n-2)2+9n-1=(5*3n-2)2 → (4*3n-1)2+9n=(5*3n-1)2
Ïðèìåð ¹3: n=5
(4*34)2+95=(5*34)2 → 3242+ 95= 4052
è ñîîòâåòñòâåííî â îáùåì âèäå
Ôîðìóëà¹2 (k (y2–1)/2(√(ky))n-2)2+(ky)n =(k((y2–1)/2+1)(√(ky))n-2)2
Ïðèìåð ¹4: n=3 k=2 y=5
(2(52–1)/2 √10)2+103=(2(52–1)/2+1) √10)2 → 242*10 +10 3=262*10
Ïðèìåð ¹5: n=4 k=3 y=7
(3 (72–1)/2 (√3*7) 2) 2+(3*7)4=((3(72–1)/2+1) (√3*7)2)2 →
15122 +214=15752
Ïðàêòè÷åñêîå çíà÷åíèå èìåþò ôîðìóëû ¹1 è ¹2, òàê êàê áåç îñîáûõ àðèôìåòè÷åñêèõ óñèëèé ðåøàþòñÿ óðàâíåíèÿ x2+y2=z2 è x2+yn=z2, ïðè ýòîì êîýôôèöèåíò k ìîæåò èìåòü ëþáûå ïîëîæèòåëüíûå çíà÷åíèÿ, â òîì ÷èñëå è èððàöèîíàëüíûå.
Ïîñòóïèëà â ðåäàêöèþ 20.08.2014 ã.