1.Introduction

The following result is known according to A111152 of The On-Line
Encyclopedia of Integer Sequences. 

A111152: Smallest number that is a sum of two n-th powers of positive
rationals but not of two n-th powers of positive integers.

a(3) = 6 = (17/21)^3 + (37/21)^3 

a(4) = 5906 = (25/17)^4 + (149/17)^4 

a(5) = 68101 = (15/2)^5 + (17/2)^5 

a(6) <= 164634913 = (44/5)^6 + (117/5)^6 (John W. Layman, Oct 20, 2005) 

a(7) <= 69071941639 = (63/2)^7 + (65/2)^7 





2. Search results

I searched for a(n) under the condition  8 <= n <= 20.
Search results.

a(8)  <= (50429/17)^8 + (43975/17)^8  
       = 8000587738704025541501346146

a(9)  <= (257/2)^9 + (255/2)^9        
       = 18456877714042519561

a(10) <= (1199/5)^10 + (718/5)^10     
       = 632498552177152162935401

a(11) <= (1025/2)^11 + (1023/2)^11    
       = 1267717091528772810596116594699

a(12) <= (9298423/17)^12 + (8189146/17)^12
       = 873135263681497645296811652793869145886016236198018083488332176234017

a(13) <= (4097/2)^13 + (4095/2)^13
       = 22300848878370942509275085017841058308947981

a(14) <= (76443/5)^14 + (16124/5)^14
       = 38119575480802651879586623497942915299371330585552625864257

a(15) <= (16385/2)^15 + (16383/2)^15
       =100433667051352806208393809176059166152175097085205816541199
       
a(16) <= (3294416782861362/97)^16 + (2731979866522411/97)^16
       = 329085218068196428341859414810924517877548890557468619562806745469579088545392836353950326748230876
         6739178243095205940051442851813420295698422427277104971904437749198746053291898220955688613607044419866444222098041537
         
a(17) <= (65537/2)^17 + (65535/2)^17
       =115792092903868957095560145391246449496637440592185117011498047040790665887761

a(18) <= (1721764/5)^18 + (922077/5)^18
       =4635225195566536797748907596675479803091770705977354080474498981834393826843075294632824262306748849

a(19) <= (262145/2)^19 + (262143/2)^19
       =34175792659776834772444496727880379705776829800698789419999532161154069279203269865191990697656339

a(20) <= (726388197629/17)^20 + (86503985645/17)^20
       = 4115258828843061827705668244865450737652683803097254856667379730951145107205028686018211
         97065153382556541604179071600181060311208139388793045411101864754465638110967289659148931275593182508930836017397200018787426
       
Case 1. n is odd (n=5,7,9,11,13,15,17,19)

       x^n + y^n = (x + y)*(x^(n-1)+.....+y^(n-1)) = k
       Take x = a/c, y = b/c
       
       a^n + b^n = (a+b)*(a^(n-1)+...+ b^(n-1)) = c^n*k.

       Assume that a,b are odd and c is even where gcd(a,b)=1.
       Set a = p + q, b= p - q.
       Then, a^n + b^n = 2*p*f(p,q) = c^n*k.
       Here p and q are opposite parities and they are relative prime.
       So,f(p,q) is odd and c is even then 2*p is divisible by c^n.
       We can get a solution when 2*p = c^n.
       Solution of 2*p = c^n is p = 2^(n-1), c = 2. 
       a = 2^(n-1) + q, b = 2^(n-1) - q.
       In general, a^n + b^n is minimized at a = b where a + b = const.
       So,when q = 1, a and b have the nearest value.
       Therefore, a = 2^(n-1) + 1, b = 2^(n-1) -1.

       Consequently, we get a smallest solution, x = ( 2^(n-1) + 1 )/2 ,
       y = ( 2^(n-1) - 1 )/2.

       For example

       n   2^(n-1)    x^n + y^n

       5   16         (17/2)^5 + (15/2)^5
       7   64         (65/2)^7 + (63/2)^7
       9   256        (257/2)^9 + (255/2)^9
      11   1024       (1025/2)^11 + (1023/2)^11
      13   4096       (4097/2)^13 + (4095/2)^13
      15   16384      (16385/2)^15 + (16383/2)^15
      17   65536      (65537/2)^17 + (65535/2)^17
      19   262144     (262145/2)^19 + (262143/2)^19


       Let x = ( 2^(n-1) + q )/2, y = ( 2^(n-1) - q )/2.
       x^n + y^n is minimized at q = 1.
       For example, n = 7, 2^(n-1)=64.

       q   x^7 + y^7

       1   (65/2)^7 +  (63/2)^7 = 69071941639(smallest)
       3   (67/2)^7 +  (61/2)^7 = 71901987823
       5   (69/2)^7 +  (59/2)^7 = 77617224511
       7   (71/2)^7 +  (57/2)^7 = 86328262903
       9   (73/2)^7 +  (55/2)^7 = 98201826199


Case 2. n is even (n=2*m,m is odd) (n=6,10,14,18)

        x^n + y^n = (x^2 + y^2)*(x^(n-2)+.....+y^(n-2)) = k.
        Take x = a/c, y = b/c.
       
        a^n + b^n = (a^2 + b^2)*f(a,b) = c^n*k.
        Here c is odd and a and b are opposite parities.
        In general, odd prime factors of a^2 + b^2 are expressed with
        4*h+1.
        So, c must be divisible by prime expressed with 4*h+1.
        5 is the smallest odd prime number expressed with 4*h+1.

        Let x^n + y^n = f(c) = (a/c)^n + (sqrt(c^n-a^2)/c)^n.
        Since f(c) is an increasing function about c,
        so x^n + y^n is minimized at c is smallest.
        Therefore,c becomes equal to 5 where a^2 + b^2 = c^n.

        Set c = 5.
        If 5 does not divide m then f(a,b) is not divisible by 5 where
        a and b are opposite parities.
        So,a^2 + b^2 must be divisible by 5^n.
        If we solve the equation a^2 + b^2 = 5^n,we can get a smallest
        solution x = a/5, y = b/5.

        If 5 divides m then 2*p and f(a,b) are not relaively prime.
        If we solve the equation a^2 + b^2 = 5^(n-1),we can get a smallest
        solution x = a/5, y = b/5.


        For example, n = 7.

        n=6,       44^2 + 117^2 = 5^6
                  (44/5)^6 + (117/5)^6     = 164634913

                   2035^2 + 828^2 = 13^6
                  (2035/13)^6 + (828/13)^6 = 14780612321281

        n=10,      1199^2 + 718^2 = 5^9 ( not 5^10)
                  (1199/5)^10 + (718/5)^10

                   341525^2 + 145668^2 = 13^10
                  (341525/13)^10 + (145668/13)^10

        n=14,      76443^2 + 16124^2 = 5^14
                  (76443/5)^14 + (16124/5)^14

                   58317492^2 + 23161315^2 = 13^14
                  (58317492/13)^14 + ( 23161315/13)^14

        n=18,      1721764^2 + 922077^2 = 5^18
                  (1721764/5)^18 + (922077/5)^18

                   4241902555^2 + 9719139348^2 = 13^18
                  (4241902555/13)^18 + (9719139348/13)^18

Case 3. n is even (n=2*m,m is even) (n=4,8,12,16,20)

        The case of n=8, (50429/17)^8 + (43975/17)^8 was found.
        17 is the first prime number which is shown by 16*h+1.

        The case of n=12, (9298423/17)^12 + (8189146/17)^12 was found.
        x^12 + y^12 = (x^4 + y^4)*(x^8 - x^4*y^4 + y^8). 
        17 is the first prime number which is shown by 8*h+1.
        (9298423/17)^4 + (8189146/17)^4 = 0 mod 17^12.

        The case of n=16,(3294416782861362/97)^16+(2731979866522411/97)^16
        was found.
        But,I don't know whether the solution for the case of n=16 is a smallest one.
        
        
        The case of n=20, (726388197629/17)^20 + (86503985645/17)^20
        was found.
        x^20 + y^20 = (x^4 + y^4)*(x^16-x^12*y^4+x^8*y^8-x^4*y^12+y^16).
        17 is the first prime number which is shown by 8*h+1.
        (726388197629)^4 + (86503985645)^4 = 0 mod 17^20.

        For example, n = 8.

        (50429/17)^8 + (43975/17)^8 =  8000587738704025541501346146
        (55927/17)^8 + (6934/17)^8  =  13720844539329207715755100097
        (103029/17)^8 + (40480/17)^8 = 1821088525780747242317450154721
        (127679/17)^8 + (23788/17)^8 = 10124300802985725813042901077377

        (27738319/97)^8 + (19395769/97)^8 = 47271978110517337561634786256230201358212162

                     n = 12
                     
        (9298423/17)^12 + (8189146/17)^12 =  873135263681497645296811652793869145886016236198018083488332176234017
        (62815285/17)^12 + (178389/17)^12 =  6477343938528555163082271072512151694296215518144755706299553816650285439633186
        (64910572/17)^12 + (5491263/17)^12 = 9602806503440700166467054748395284811061145474333148450066067639543259245670977

                     n = 16

        (3294416782861362/97)^16 + (2731979866522411/97)^16
        = 329085218068196428341859414810924517877548890557468619562806745469579088545392836353950326748230876673917824309520
          5940051442851813420295698422427277104971904437749198746053291898220955688613607044419866444222098041537
 
        (3541162157737595/97)^16 + (2753420643750823/97)^16
        = 101313865557141380022875594072051170556086704401596494786522475825521918882345808637874725138247773729168474125590
          82941716092812365581088230055445656896065613235617369184534276666901294054739509843343946300176605228866






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