1.Introduction


Choudhry gave a simple and beautiful identity.([1])

(a-a^3-2a^5+a^9)^n  +  (1+a^2-2a^6+2a^7+a^8)^n +  (2a^3+2a^4-2a^7)^n =

(a+3a^3-2a^5+a^9)^n + (1+a^2-2a^6-2a^7+a^8)^n + (-2a^3+2a^4+2a^7)^n,for n=1,5.  

Though I couldn't find above identity by my method,two identities were obtained. 

I show them as follows.



2. Solutions

         

         1. x1^n + x2^n + x3^n = y1^n + y2^n + y3^n, (n=1,5)

           p,q:arbitrary

           x1 = 5p^7 + 6qp^6 + 51q^2p^5 + 58q^3p^4 + 108q^4p^3 + 96q^5p^2 + 36q^6p + 40q^7 
           x2 = 5p^7 - 10qp^6 + 39q^2p^5 - 90q^3p^4 + 52q^4p^3 - 160q^5p^2 + 4q^6p - 40q^7
           x3 = -p(5p^6 - 4qp^5 + 39q^2p^4 - 32q^3p^3 + 52q^4p^2 - 64q^5p + 4q^6)

           y1 = -p(5p^6 + 4qp^5 + 39q^2p^4 + 32q^3p^3 + 52q^4p^2 + 64q^5p + 4q^6)
           y2 = 5p^7 - 6qp^6 + 51q^2p^5 - 58q^3p^4 + 108q^4p^3 - 96q^5p^2 + 36q^6p - 40q^7
           y3 = 5p^7 + 10qp^6 + 39q^2p^5 + 90q^3p^4 + 52q^4p^3 + 160q^5p^2 + 4q^6p + 40q^7
           
                     
           Let {x1, x2, x3, x4} = {a+b+c, a-b-c, -a+b-c, -a-b+c},
               {y1, y2, y3, y4} = {d+e+f, -d+e-f, -d-e+f,  d-e-f}.

           Condition: a+b-c = -d+e+f and c = f

           x1^5 + x2^5 + x3^5 + x4^5 -(y1^5 + y2^5 + y3^5 + y4^5)

           = 80abc(a^2+b^2+c^2) - 80def(d^2+e^2+f^2)...............................(1)

           Substitute a = x+p, b = mx+q, d = x-p, e = mx-q to (1),then (1) becomes to (2).
           
           (320q+640mp+480m^2q+160pm^3)x^3+(320pmq+320q^2)x^2
          +(640pmq^2+320q^3+160p^3m+480qp^2)x......................................(2)

           Decide m to 320q+640mp+480m^2q+160pm^3=0,then m = -q(2q^2+3p^2)/(p(4q^2+p^2)).

           We obtain x = -2/5(16q^4+8q^2p^2+p^4)qp^2/(4q^6+16q^4p^2+9q^2p^4+p^6).
            
           Since x4 equals to y4, we obtain above identity.

           This solution is same as Choudhry's one([2]).
           



         2. x1^n + x2^n + x3^n = y1^n + y2^n + y3^n, (n=1,5)
            
            x1 = 74q^4p^4 + 20q^2p^6 + 29q^3p^5 + qp^7 + 84p^3q^5 + 64q^6p^2 + 36pq^7 + 2p^8 + 40q^8
            x2 = q(-74q^3p^4 - 8qp^6 + 45p^5q^2 + 7p^7 + 44p^3q^4 - 128q^5p^2 + 4pq^6 - 40q^7)
            x3 = -p(12p^5q^2 + 41q^3p^4 + 3qp^6 + 84q^5p^2 - 64pq^6 - 28q^7 + 2p^7)

            y1 = q(74q^3p^4 + 8qp^6 + 45p^5q^2 + 7p^7 + 44p^3q^4 + 128q^5p^2 + 4pq^6 + 40q^7)
            y2 = -74q^4p^4 - 20q^2p^6 + 29q^3p^5 + qp^7 + 84p^3q^5 - 64q^6p^2 + 36pq^7 - 2p^8 - 40q^8
            y3 = p(12p^5q^2 - 41p^4q^3 - 3p^6q - 84p^2q^5 - 64pq^6 + 28q^7 + 2p^7)

            Condition: a+b-c = -d+e+f and ab = de

            x1^5 + x2^5 + x3^5 + x4^5 -(y1^5 + y2^5 + y3^5 + y4^5)

            = 80abc(a^2+b^2+c^2) - 80def(d^2+e^2+f^2)............................................(3)

            Substitute a = p(x+1), b = q(x-1), d = q(x+1), e = p(x-1) to (3),then (3) becomes to (4).
           
            (320qp^2-160p^3+640mq^2+160mp^2-640pq^2-320q^3)x^2+(-320pq^2+960pqm-960qp^2+320p^3)x
            -160pq^2+1120mp^2+160mq^2-480pm^2+160m^3-800p^3......................................(4)

            Decide m to (320qp^2-160p^3+640mq^2+160mp^2-640pq^2-320q^3)=0,then m = (-2qp^2+p^3+4pq^2+2q^3)/(4q^2+p^2).

            We obtain x = (4p^6+37q^2p^4+64q^4p^2+20q^6)q/((p^2-2q^2)p(4q^2+p^2)^2).
            
            Since x4 equals to y4, we obtain above identity.


3. Example

      Case 1.   

      (p,q)       x1       x2       x3       y1       y2       y3
      
      (1 1)        2       -1        0       -1        0        2
      (1 2)    12909    -9731     1063    -3561    -4427    12229
      (1 3)    18481   -15167     1118    -2989    -9617    17038
      (1 4)    37333   -31787     1491    -3917   -23259    34213
      (1 5)   101589   -88508     2701    -7500   -70025    93307
      (2 1)      618      -56     -161     -417      184      634
      (2 3)     1724    -1162      127     -641     -346     1676
      (2 5)   153734  -122284    11211   -32053   -68412   143126
      (3 1)     4541      721    -1968    -3489     2047     4736
      (3 2)     7727    -2081    -1083    -4683     1399     7847
      (3 4)  3256351 -2054497   196233 -1338519  -460177  3196783
      (3 5)   256622  -181496    20643   -86286   -65084   247139
      (4 1)      911      293     -498     -754      501      959
      (4 3)   269017   -91493   -25426  -155218    35179   272137
      (4 5)   446535  -269987    22390  -192906   -48659   440503
      (5 1)    17403     7562   -10873   -15078    10781    18389
      (5 2)   363013    17557  -129441  -263921   139245   375805
      (5 3)   124510   -25460   -22807   -78622    27976   126889
      (5 4)  2588373  -977963  -183629 -1450989   265189  2612581




      Case 2.

      (p,q)       x1       x2       x3       y1       y2       y3      

      (1 1)        7       -3       -1        7       -1       -3
      (1 2)    11566    -8677     2304    10971    -4036    -1742
      (1 3)   207601  -170571    43127   191247  -107671    -3419
      (1 4)    71611   -61026    12641    65566   -44503     2163
      (1 5)  9875191 -8607895  1459779  9063555 -6796561   460081
      (2 1)      634       56     -417      618     -184     -161
      (2 3)     4516    -2958      509     4404    -1054    -1283
      (2 5)  3529894 -2808540   761171  3285910 -1569484  -233901
      (3 1)    22961     7213   -20001    20231   -11351     1293
      (3 2)     7232     -911    -3318     7217    -1346    -2868
      (3 4)  5656631 -3396806   298077  5571002 -1104971 -1908129
      (3 5)   550801  -383215    84939   531835  -150391  -128919
      (4 1)     1423      549    -1378     1087     -907      414
      (4 3)   735499  -159663  -271174   736539  -117439  -314438
      (4 5)  9685999 -5487595   137046  9583455 -1725019 -3522986
      (5 1)   132107    51853  -134001    86751   -96221    59429
      (5 2)   412418    94227  -324068   385235  -164360   -38298
      (5 3)  1439525   -64875  -771017  1428423  -314243  -510547
      (5 4)  4660991 -1242146 -1482339  4671582  -698267 -2036809

4.References

      1. Tito Piezas:http://sites.google.com/site/tpiezas/020
   
      2. Ajai Choudhry:On equal sums of fifth powers,Indian J. pure appl. Math., 28(1997)