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.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 + 175967^21. Search resultsLet 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.2. Method(elliptic curve)