PROPOSED RESTATEMENT OF PYTHAGOREAN THEOREM
It is being observed that the original Theorem postulated by Pythagoras needs a bit of restatement in the following manner:
The actual representation should have been X2 +YN =Z2 and ‘N’ can be any digit either even digit or odd digit and value of ‘N ’may be made infinitely long.
When value of ‘N’ is an even digit i.e., divisible by two then the present form of Pythagorean Triplets can hold good and can be expressed as X2 + (Y(N/2))2 = Z2
But when value of ‘N’ is an odd digit odd then let’s see what happens.
In the situations where ‘N’ is an odd digit ‘N’ can become either a prime digit or a non-prime (e.g., in the first case let N be 29 and in the second case N can become say 27)
It can be proved that there may be infinite examples of the identity XN + Y2 = Z2 may be found when ‘N’ can be anything like 7, 11, 15, 17, 19, 29, 77, 101, 199, 1999, 19999 … 7165391 etc and may be said → ∞ but the other two exponents in the triplets are always 2.
The method of generation of such types of identities are yet to be found in any books, periodicals, journals and even in Encyclopaedia of Mathematics till date.
It can be proposed that the in the most generalised form of Diophantine Triplets is:
XN + Y2 – Z2 =0 or,
XN = Z2 – Y2
and to prove the veracity of my claim I beg to submit only 10 such examples although there are infinite examples and after checking these examples if the proposition satisfies you I can forward several more examples which you can check for correctness and a video demonstrating the methodology of generation of such examples.
Example 1:
(4799539515572734)2 + (10197)7 = (4800733567869263)2
May be expressed as:
(4800733567869263)2 - (4799539515572734)2 = (10197)7
Example 2:
(256821779852045695)2 = (243399120541893294)2 + (1189)11
May also be expressed as:
(256821779852045695)2 - (243399120541893294)2 = (1189)11
EXAMPLE 3:
(72442039641464233398437)2 + (35)15 = (72442039641464233398438)2
Alternatively:
(72442039641464233398438)2 - (72442039641464233398437)2 = (35)15
Example 4:
(468594620818201364526055557435883)2 + (87)17 = (468594620818201364526055557435884)2
May also be expressed as:
(468594620818201364526055557435884)2 - (468594620818201364526055557435883)2 = (87)17
Example 5:
(8445026930272407151927071611)2 = (8445026880290893598026364318)2 + (333)19
Alternatively, being expressed as:
(8445026930272407151927071611)2 - (-8445026880290893598026364318)2 = (333)19
Example 6:
(598073737843332430390525052102969984)2
+ (-598073737843332430390524934714561817)2
= (303)19
Example No.7:
(-7751445471943785321966543108315)2 + (91)29 = (7751487330718610893302991997206)2
Alternatively, being expressed as:
(7751487330718610893302991997206)2 - (7751487330718610893302991997206)2 = (91)29
Example 8:
(1811327378603031639774565721195023429740931634232114361138243907589640002697706222534179687)2 + (15)77
= (1811327378603031639774565721195023429740931634232114361138243907589640002697706222534179688)2
Or,
(1811327378603031639774565721195023429740931634232114361138243907589640002697706222534179688)2 - (1811327378603031639774565721195023429740931634232114361138243907589640002697706222534179687)2 = (15)77
Example 9:
(1063791482518319402684855428336457802756523246811292807437191173592935754373316278748254215121313471465709294197320200625122654733661481920967955428211003319571036195246925871244171454220791994747)2 + (91)101
= (1063791482518319402684855428336457802756523246811292807437191173592935754373316278748254215121313471465709294197320200625122654733661481920967955428211003319571036195246925871244171454220791995090)2
Or,
(1063791482518319402684855428336457802756523246811292807437191173592935754373316278748254215121313471465709294197320200625122654733661481920967955428211003319571036195246925871244171454220791995090)2 - (1063791482518319402684855428336457802756523246811292807437191173592935754373316278748254215121313471465709294197320200625122654733661481920967955428211003319571036195246925871244171454220791994747)2 = (91)101
Example 10:
-(-1617238254812378995672323884550108405428601599452312691593393464098239919341541592368)2 + (14)101
= (1617238254812378995672323884550108405428601599452312709340501867293451540295386467632)2
Or,
(1617238254812378995672323884550108405428601599452312709340501867293451540295386467632)2 - (1617238254812378995672323884550108405428601599452312691593393464098239919341541592368)2 = (14)101
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