Find the smallest N that is the sum of a square and fourth power in n different ways.
N=a^2+b^4.
For example, 17 is the sum of a square and fourth power in two different ways.
17=1^4 + 4^2 = 2^4 + 1^2.
First, we looked for the integer solutions by brute-force.
When there is no solution by the brute-force method, we searched for it by elliptic curve method.

2018.4.4: Add 128198143090625 and 3234825286225.

2018.3.28: Add 205117028945.

1. Search results 

   Smallest: N
   Search range: N<2*10^9(brute-force)
   
   Table of smallest solutions  of N=a^2+b^4.
   
   17 = 1^4 + 4^2
      = 2^4 + 1^2
      
   3026 = 1^4 + 55^2
        = 5^4 + 49^2
        = 7^4 + 25^2

   141457 =  3^4 + 376^2
          = 13^4 + 336^2
          = 14^4 + 321^2
          = 18^4 + 191^2

   4740625 =  6^4 + 2177^2
           = 10^4 + 2175^2
           = 35^4 + 1800^2
           = 45^4 +  800^2
           = 46^4 +  513^2
        
   113260225 = 15^4 + 10640^2
             = 32^4 + 10593^2
             = 49^4 + 10368^2
             = 60^4 + 10015^2
             = 84^4 +  7967^2
             = 95^4 +  5640^2

   205117028945 =  34^4 + 452897^2
                = 337^4 + 438428^2
                = 422^4 + 416417^2
                = 433^4 + 412268^2
                = 527^4 + 357748^2
                = 593^4 + 285412^2
                = 646^4 + 175967^2

2. Method(elliptic curve) 

Let y^2=-x^4+N........................................(1)

Find the rational solutions of equation (1) for some natural number N.
Let the rational solutions (x,y)={(a1/d1,c1/d1^2),(a2/d2,c2/d2^2),....,(an/dn,cn/dn^2)}.
Multiply (x,y) by d1*d2...dn, we obtain an integer solution of Y^2=-X^4+N*(d1*d2...dn)^4.
In this way, we can obtain the solutions of y^2=-x^4+N in n different ways for arbitrary n.

Example1:
y^2=-x^4+17...........................................(2)
This quratic curve is birationally equivalent to an elliptic curve below.
Elliptic curve V^2=U^3+68U has rank 2, then there are infinitely many rational solutions.
Small seven solutions of equation (2) are  (x,y)={(1, 4),(2, 1),(2/5, 103/25),(19/13, 596/169),
(26/29, 3401/841),(43/85, 29732/7225),(191/97, 13196/9409)}.

1587000129244572305637310625 = 6216730^4+ 9661932973225^2
                             = 3108365^4+ 38647731892900^2
                             = 1243346^4+ 39807163849687^2
                             = 4542995^4+ 34074035810900^2
                             = 2786810^4+ 39072810989225^2
                             = 1572467^4+ 39760358638052^2
                             = 6120595^4+ 13550735201900^2

However 1587000129244572305637310625 is not a smallest number in seven different ways.

Example2:
y^2=-x^4+205117028945..................................(3)
This quratic curve is birationally equivalent to an elliptic curve below.
Elliptic curve V^2=U^3+820468115780U has rank 6, then there are infinitely many rational solutions.
Small ten solutions of equation (3) are  (x,y)={(646, 175967),(593, 285412),(527, 357748),(433, 412268),
(422, 416417),(337, 438428),(34, 452897),(779/5, 11306188/25),(3349/5, 1550468/25),(2578/5, 9166663/25)}.

128198143090625 = 3230^4+ 4399175^2
                = 2965^4+ 7135300^2
                = 2635^4+ 8943700^2
                = 2165^4+ 10306700^2
                = 2110^4+ 10410425^2
                = 1685^4+ 10960700^2
                =  170^4+ 11322425^2
                =  779^4+ 11306188^2
                = 3349^4+ 1550468^2
                = 2578^4+ 9166663^2
                            
128198143090625 may be a smallest number in ten different ways.

Example3:
y^2=-x^4+113260225..................................(4)
This quratic curve is birationally equivalent to an elliptic curve below.
Elliptic curve V^2=U^3+453040900U has rank 6, then there are infinitely many rational solutions.
Small eight solutions of equation (4) are  (x,y)={(95, 5640), (60, 10015), (49, 10368), (32, 10593), (15, 10640),
(84, 7967), (1341/13, 31792/169), (979/13, 1521912/169)}.

3234825286225 = 1235^4+ 953160^2
              =  780^4+ 1692535^2
              =  637^4+ 1752192^2
              =  416^4+ 1790217^2
              =  195^4+ 1798160^2
              = 1092^4+ 1346423^2
              = 1341^4+ 31792^2
              =  979^4+ 1521912^2
                            
3234825286225 may be a smallest number in eight different ways.