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Palindromic Pythagorean Triples
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Introduction

Palindromic numbers are numbers which read the same from
 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random ( ? ) examples : 4004, 630036, 1559551

Palindromic Pythagorean Triples are defined and calculated by this extraordinary intricate and excruciatingly complex formula.
So, this line is for experts only

pal_X 2  +  pal_Y 2  =  pal_Z 2



Sources Revealed

Here are a few general websources that will teach you how to construct pythagorean triples.

blue Pythagorean Theorem
blue Pythagorean triple - Eric W. Weisstein

Finding Palindromic Pythagorean Triples is quite another matter.
I'm counting on you to find them !


[ January 23, 2005 ]
Zakir Seidov contributed a list of palindromic areas of  primitive pythagorean triangles.
I have displayed them in a separate wonplate. Please find next the reference address ¬
won163.htm


Research Results

Of course, the well known triplet (3,4,5) really was not too hard to find, was it !

32  +  42  =  52
Neither was one of its multiples with 11, 101, 111, 1111, etc. :
3032  +  4042  =  5052

John Abreu sent me a method to find more palindromic multiples of the above pythagorean triplet (3,4,5).

Let S be any finite sequence of 0's and 1's, let S' be its reflection; for example, if S=101011, then S'=110101.
Define X:= SS', then X is a palindrome containing only 0's and 1's
( in the example X := 101011110101).
Besides, if A is any digit from 0 through 9, the number AX := A.X is also a palindrome.
Now, multiply the identity 3^2 + 4^2 = 5^2 by X^2 to obtain:
(3X)^2 + (4X)^2 = (5X)^2
The last identity is telling us that the palindromes 3X, 4X and 5X form a pythagorean triple.


The real challenge is to find triplets where the palindromic numbers have NO COMMON factors.
For the moment I can only supply NEAR MISSES !

202  +  992  =  1012
2522  +  2752  =  3732
3632  +  4842  =  6052


[ August 29, 1998 ]
Jud McCranie wrote me the following email ¬

" One of your pages asks for Pythagorean Triples that are palindromes, other than ones based on 3-4-5.
I did a search and I didn't find any. Here are ones where both legs are palindromes. In several cases,
the third number is a "near miss" - changing one digit by one, or exchanging two consecutive digits would
result in a palindrome."
              3              4              5
              6              8             10
            363            484            605
            464            777            905
           3993           6776           7865
           6776          23232          24200
            313          48984          48985
           8228          69696          70180
          30603          40804          51005  close
          34743          42824          55145  close
          29192          60006          66730
          25652          55755          61373
          52625          80808          96433
          36663         616616         617705
          48984         886688         888040
         575575        2152512        2228137
           6336        2509052        2509060
        2327232        4728274        5269970
        3006003        4008004        5010005  close
        3458543        4228224        5462545  close
          80308        5578755        5579333
        2532352        5853585        6377873
        5679765       23711732       24382493
        4454544       29055092       29394580
         677776      237282732      237283700
      300060003      400080004      500100005  close
      304070403      402080204      504110405  close
      276626672      458515854      535498930
      341484143      420282024      541524145  close
      345696543      422282224      545736545  close
      359575953      401141104      538710545
      277373772      694808496      748127700
      635191536     2566776652     2644203220
     6521771256    29986068992    30687095560
    21757175712    48337273384    53008175720 
    30000600003    40000800004    50001000005  close
    30441814403    40220802204    50442214405  close
    34104840143    42002820024    54105240145  close
    27280108272    55873637855    62177710753



Contributions

John Abreu (email) gives a method to find more palindromic multiples of the triplet (3,4,5) - go to topic

Jud McCranie (email) made an attempt to find palindromic triples - go to topic




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