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**FERMAT’S LAST THEOREM: THE PROOF RESULTING IN CALCULATING ALGORITHM.**

* *

*by Ivan M.
Obushenko, Ph.D.*

E-mail: obush@estart.com

*Printed on March 8-10, 2005*

The new approach for consideration of the legendary Fermat’s Last Theorem has been developed. One advantage has been received from proposed consideration: the calculating algorithm for PC – “ Obushenko-Fermat MachineÔ ” making the calculation process by fully automated and expendable unlimitedly . The “byproduct” of calculation is a finding of all integer triads fulfilling to the Pythagoras Theorem.

For integers *a, b,
c, *and *n*>2 a^{n} + b^{n }¹ c^{n}
.

The current consideration is substantially based on following Lemma expressing my basic idea led me forward.

__Lemma__*.* For any three
numbers *a, b, c*, for which * a *+* b **> c* some triangle can be constructed.

That *Lemma* arisen from the everyday experience and
can’t be proved in principle on my opinion. So it rather *Axiom* than *Lemma*
. Indeed, let’s make a drawing for *a, b **< c* :

*a m b
*

*Drawing 1.*

*c*

* *

Case#1. __What,
if a + b ____<____ c__

__ __

Let’s put for clarity *|a|<|b|<|c|, where |a|*=mod*
a* .

** ***b’ Drawing. 2*

*a b*

* *

*c*

c=900 a=252 b=864; c=900 a=540 b=720

c=901 a=60 b=899; c=901 a=424 b=795; c=901 a=451 b=780; c=901 a=476 b=765

c=902 a=198 b=880

c=904 a=120 b=896

c=905 a=95 b=900; c=905 a=464 b=777; c=905 a=543 b=724; c=905 a=616 b=663

c=909 a=180 b=891

c=910 a=224 b=882; c=910 a=350 b=840; c=910 a=462 b=784; c=910 a=546 b=728

c=914 a=336 b=850

c=915 a=165 b=900; c=915 a=408 b=819; c=915 a=549 b=732; c=915 a=621 b=672

c=916 a=240 b=884

c=918 a=432 b=810

c=920 a=552 b=736

c=922 a=522 b=760

c=923 a=355 b=852

c=925 a=43 b=924; c=925 a=259 b=888; c=925 a=285 b=880; c=925 a=300 b=875; c=925 a=520 b=765; c=925 a=533 b=756; c=925 a=555 b=740

c=928 a=640 b=672

c=929 a=129 b=920

c=930 a=558 b=744

c=932 a=420 b=832

c=935 a=143 b=924; c=935 a=396 b=847; c=935 a=440 b=825; c=935 a=561 b=748

c=936 a=360 b=864

c=937 a=215 b=912

c=939 a=75 b=936

c=940 a=564 b=752

c=941 a=580 b=741

c=942 a=510 b=792

c=943 a=207 b=920

c=945 a=567 b=756

c=949 a=301 b=900; c=949 a=365 b=876; c=949 a=420 b=851; c=949 a=624 b=715

c=950 a=266 b=912; c=950 a=570 b=760

c=951 a=225 b=924

c=952 a=448 b=840

c=953 a=615 b=728

c=954 a=504 b=810

c=955 a=573 b=764

c=957 a=660 b=693

c=959 a=616 b=735

c=960 a=576 b=768 <

c=962 a=62 b=960; c=962 a=312 b=910; c=962 a=370 b=888; c=962 a=638 b=720

c=964 a=480 b=836

c=965 a=124 b=957; c=965 a=387 b=884; c=965 a=475 b=840; c=965 a=579 b=772

c=969 a=456 b=855

c=970 a=88 b=966; c=970 a=186 b=952; c=970 a=582 b=776; c=970 a=650 b=720

c=975 a=108 b=969; c=975 a=240 b=945; c=975 a=273 b=936; c=975 a=375 b=900; c=975 a=495 b=840; c=975 a=585 b=780; c=975 a=612 b=759

c=976 a=176 b=960

c=977 a=248 b=945

c=979 a=429 b=880

c=980 a=588 b=784

c=981 a=540 b=819

c=984 a=216 b=960

c=985 a=140 b=975; c=985 a=473 b=864; c=985 a=591 b=788; c=985 a=696 b=697

c=986 a=264 b=950; c=986 a=310 b=936; c=986 a=464 b=870; c=986 a=680b=714

c=988 a=380 b=912

c=990 a=594 b=792

c=995 a=597 b=796

c=997 a=372 b=925

c=999 a=324 b=945

c=1000 a=280 b=960; c=1000 a=352 b=936; c=1000 a=600 b=800.