New Conjectures Related to Sum of Powers Francisco Duque Lemos franciscoduquelemos@ist.utl.pt INTRODUCTION The next conjectures I propose are based on computational work about equations with sum of powers. REASON & MOTIVATIONS FOR THE CONJECTURES The logic or the motive for the following conjectures I suggest is related to the Fermat Last Theorem. In the end, such conjectures result in an extended version of the abc conjecture. Fermat Last Theorem:

+

=

(1) It has been proven that with n > 2 (n being an integer) and A, B and C positive integers,thereisnosolution.Andthatshouldbetheendofthetheorem,butfaraway from it. Corollary: Squaring equation (1) we obtain:

2+2+2()

= 2 That has no positive integer solution with n > 2. Which means that:

+

+

=

With n > 4 and =2

()0,5 there are no positive integer solutions, but what about other instances?? CONJECTURES CITATION Conjecture 1:

=

( 2)

=1 Theseequationshavenosolutionwithinthenaturalnumbersdomain,withk> 2(n-1). At the limit of the conjecture exists: 958004+2175194+4145604= 4224814 (given by Roger Frye, disproving the Euler Conjecture about the sum of powers) Conjecture 2:

=

+1

=1 The equations that are solutions of this equality, with all k > 2(n-1), imply that all

and have a common prime factor. Computational work developed on the equation

+

+

=

give the results that are showed in the annex. At the limit of this 3rd conjecture (with n =3) exists: 1m + 26 + 24 = 34 The 2nd conjecture (with n = 2) is the Beal conjecture. [1] The case n = 1 had been already proven. Conjecture 3:

=

+1

=1( 2) There are a finite number of solutions to each n where the bases are coprime and the exponents confirm the equation1

1+1=1. (Fermat-Catalan conjecture extension) The conjecture 2 limits the existence of coprime solutions to k > 2(n-1). Conjecture 4:

= ( 2)

=1 With the last 3 conjectures I can suppose this conjecture. For each > 0 and > 1 + there are only a finite number of solutions where their elements are co-primes. (ABC conjecture extension) ANNEX I have used parallel computation with 360 CPUs to solve the algorithms of these equations but, due to the lack of resources, the focus of the search was on the case of n = 3, bases from 1 to 1000 and exponents from 5 to 15. All the equations found confirm the conjectures: Similar algorithms had been applied to the other values of n and no counter-example was found within 2 years of computational work. 10^6 + 5^8 + 10^7 = 15^6 10^7 + 5^8 + 10^6 = 15^6 10^15 + 5^16 + 250^5 = 75^8 10^6 + 10^7 + 5^8 = 15^6 10^7 + 10^6 + 5^8 = 15^6 10^15 + 25^8 + 250^5 = 75^8 10^15 + 250^5 + 25^8 = 75^8 10^15 + 250^5 + 5^16 = 75^8 1000^5 + 5^16 + 250^5 = 75^8 1000^5 + 25^8 + 250^5 = 75^8 1000^5 + 250^5 + 25^8 = 75^8 1000^5 + 250^5 + 5^16 = 75^8 108^8 + 18^12 + 324^6 = 18^13 108^8 + 18^12 + 18^12 = 18^13 108^8 + 324^6 + 324^6 = 18^13 108^8 + 324^6 + 18^12 = 18^13 11^5 + 11^7 + 11^7 = 33^5 11^7 + 11^5 + 11^7 = 33^5 11^7 + 11^7 + 11^5 = 33^5 11^10 + 11^12 + 121^6 = 363^5 11^10 + 11^12 + 11^12 = 363^5 11^12 + 11^10 + 121^6 = 363^5 11^12 + 11^10 + 11^12 = 363^5 11^12 + 11^12 + 121^5 = 363^5 11^12 + 11^12 + 11^10 = 363^5 11^15 + 11^17 + 11^17 = 3993^5 11^17 + 11^15 + 11^17 = 3993^5 11^17 + 11^17 + 11^15 = 3993^5 11^10 + 121^6 + 121^6 = 363^5 11^10 + 121^6 + 11^12 = 363^5 11^12 + 121^5 + 121^6 = 363^5 11^12 + 121^5 + 11^12 = 363^5 11^12 + 121^6 + 121^5 = 363^5 11^12 + 121^6 + 11^10 = 363^5 11^20 + 121^11 + 121^11 = 43923^5 121^5 + 11^12 + 121^6 = 363^5 121^5 + 11^12 + 11^12 = 363^5 121^6 + 11^10 + 121^6 = 363^5 121^6 + 11^10 + 11^12 = 363^5 121^6 + 11^12 + 121^5 = 363^5 121^6 + 11^12 + 11^10 = 363^5 121^11 + 11^20 + 121^11 = 43923^5 121^5 + 121^6 + 121^6 = 363^5 121^5 + 121^6 + 11^12 = 363^5 121^6 + 121^5 + 121^6 = 363^5 121^6 + 121^5 + 11^12 = 363^5 121^6 + 121^6 + 121^5 = 363^5 121^6 + 121^6 + 11^10 = 363^5 121^10 + 121^11 + 121^11 = 43923^5 121^11 + 121^10 + 121^11 = 43923^5 121^11 + 121^11 + 121^10 = 43923^5 121^11 + 121^11 + 11^20 = 43923^5 121^15 + 121^16 + 121^16 = 5314683^5 121^16 + 121^15 + 121^16 = 5314683^5 121^16 + 121^16 + 121^15 = 5314683^5 122^7 + 61^8 + 244^6 = 305^6 127^8 + 127^9 + 254^7 = 508^7 127^9 + 127^8 + 254^7 = 508^7 127^8 + 254^7 + 127^9 = 508^7 127^9 + 254^7 + 127^8 = 508^7 128^5 + 4^17 + 4^17 = 64^6 128^5 + 4^17 + 4^17 = 16^9 128^5 + 4^17 + 4^17 = 8^12 128^5 + 4^17 + 4^17 = 4^18 128^7 + 8^16 + 256^6 = 1024^5 128^7 + 8^16 + 256^6 = 32^10 128^7 + 8^16 + 64^8 = 1024^5 128^7 + 8^16 + 64^8 = 32^10 128^7 + 8^16 + 16^12 = 1024^5 128^7 + 8^16 + 16^12 = 32^10 128^7 + 8^16 + 8^16 = 1024^5 128^7 + 8^16 + 8^16 = 32^10 128^7 + 16^12 + 256^6 = 1024^5 128^7 + 16^12 + 256^6 = 32^10 128^7 + 16^12 + 64^8 = 1024^5 128^7 + 16^12 + 64^8 = 32^10 128^7 + 16^12 + 16^12 = 1024^5 128^7 + 16^12 + 16^12 = 32^10 128^7 + 16^12 + 8^16 = 1024^5 128^7 + 16^12 + 8^16 = 32^10 128^9 + 16^16 + 512^7 = 8192^5 128^9 + 16^16 + 512^7 = 32^13 128^9 + 16^16 + 128^9 = 8192^5 128^9 + 16^16 + 128^9 = 32^13 128^11 + 16^19 + 16^19 = 8192^6 128^11 + 16^19 + 16^19 = 64^13 128^7 + 32^10 + 128^7 = 8^17 128^8 + 32^11 + 32^11 = 8^19 128^7 + 64^8 + 256^6 = 1024^5 128^7 + 64^8 + 256^6 = 32^10 128^7 + 64^8 + 64^8 = 1024^5 128^7 + 64^8 + 64^8 = 32^10 128^7 + 64^8 + 16^12 = 1024^5 128^7 + 64^8 + 16^12 = 32^10 128^7 + 64^8 + 8^16 = 1024^5 128^7 + 64^8 + 8^16 = 32^10 128^17 + 64^20 + 128^17 = 2048^11 128^7 + 128^7 + 32^10 = 8^17 128^9 + 128^9 + 256^8 = 8192^5 128^9 + 128^9 + 256^8 = 32^13 128^9 + 128^9 + 16^16 = 8192^5 128^9 + 128^9 + 16^16 = 32^13 128^14 + 128^14 + 512^11 = 1048576^5 128^14 + 128^14 + 512^11 = 1024^10 128^14 + 128^14 + 512^11 = 32^20 128^17 + 128^17 + 256^15 = 2048^11 128^17 + 128^17 + 64^20 = 2048^11 128^7 + 256^6 + 256^6 = 1024^5 128^7 + 256^6 + 256^6 = 32^10 128^7 + 256^6 + 64^8 = 1024^5 128^7 + 256^6 + 64^8 = 32^10 128^7 + 256^6 + 16^12 = 1024^5 128^7 + 256^6 + 16^12 = 32^10 128^7 + 256^6 + 8^16 = 1024^5 128^7 + 256^6 + 8^16 = 32^10 128^9 + 256^8 + 512^7 = 8192^5 128^9 + 256^8 + 512^7 = 32^13 128^9 + 256^8 + 128^9 = 8192^5 128^9 + 256^8 + 128^9 = 32^13 128^17 + 256^15 + 128^17 = 2048^11 128^9 + 512^7 + 256^8 = 8192^5 128^9 + 512^7 + 256^8 = 32^13 128^9 + 512^7 + 16^16 = 8192^5 128^9 + 512^7 + 16^16 = 32^13 128^14 + 512^11 + 128^14 = 1048576^5 128^14 + 512^11 + 128^14 = 1024^10 128^14 + 512^11 + 128^14 = 32^20 129^6 + 258^6 + 258^6 = 129^7 129^8 + 258^7 + 516^7 = 129^9 129^8 + 516^7 + 258^7 = 129^9 130^5 + 65^5 + 130^5 = 65^6 130^6 + 65^7 + 260^6 = 65^8 130^5 + 130^5 + 65^5 = 65^6 130^7 + 130^7 + 260^7 = 130^8 130^6 + 260^6 + 65^7 = 65^8 130^7 + 260^7 + 130^7 = 130^8 132^5 + 33^6 + 66^5 = 33^7 132^5 + 66^5 + 33^6 = 33^7 132^6 + 66^6 + 66^6 = 66^7 152^8 + 304^7 + 608^6 = 912^6 152^8 + 608^6 + 304^7 = 912^6 162^6 + 486^5 + 486^5 = 54^8 179^6 + 358^5 + 358^5 = 537^5 18^12 + 18^12 + 108^8 = 18^13 18^12 + 324^6 + 108^8 = 18^13 2^5 + 2^5 + 2^6 = 2^7 2^5 + 2^6 + 2^5 = 2^7 2^6 + 2^5 + 2^5 = 2^7 2^6 + 2^6 + 2^7 = 2^8 2^6 + 2^7 + 2^6 = 2^8 2^7 + 2^6 + 2^6 = 2^8 2^7 + 2^7 + 2^8 = 2^9 2^7 + 2^8 + 2^7 = 2^9 2^8 + 2^7 + 2^7 = 2^9 2^8 + 2^8 + 2^9 = 4^5 2^8 + 2^8 + 2^9 = 2^10 2^8 + 2^9 + 2^8 = 4^5 2^8 + 2^9 + 2^8 = 2^10 2^9 + 2^8 + 2^8 = 4^5 2^9 + 2^8 + 2^8 = 2^10 2^9 + 2^9 + 4^5 = 2^11 2^9 + 2^9 + 2^10 = 2^11 2^9 + 2^10 + 2^9 = 2^11 2^10 + 2^9 + 2^9 = 2^11 2^10 + 2^10 + 2^11 = 4^6 2^10 + 2^10 + 2^11 = 2^12 2^10 + 2^11 + 4^5 = 4^6 2^10 + 2^11 + 4^5 = 2^12 2^10 + 2^11 + 2^10 = 4^6 2^10 + 2^11 + 2^10 = 2^12 2^11 + 2^10 + 4^5 = 4^6 2^11 + 2^10 + 4^5 = 2^12 2^11 + 2^10 + 2^10 = 4^6 2^11 + 2^10 + 2^10 = 2^12 2^11 + 2^11 + 4^6 = 2^13 2^11 + 2^11 + 2^12 = 2^13 2^11 + 2^12 + 2^11 = 2^13 2^12 + 2^11 + 2^11 = 2^13 2^12 + 2^12 + 2^13 = 4^7 2^12 + 2^12 + 2^13 = 2^14 2^12 + 2^13 + 4^6 = 4^7 2^12 + 2^13 + 4^6 = 2^14 2^12 + 2^13 + 2^12 = 4^7 2^12 + 2^13 + 2^12 = 2^14 2^13 + 2^12 + 4^6 = 4^7 2^13 + 2^12 + 4^6 = 2^14 2^13 + 2^12 + 2^12 = 4^7 2^13 + 2^12 + 2^12 = 2^14 2^13 + 2^13 + 4^7 = 8^5 2^13 + 2^13 + 4^7 = 2^15 2^13 + 2^13 + 2^14 = 8^5 2^13 + 2^13 + 2^14 = 2^15 2^13 + 2^14 + 2^13 = 8^5 2^13 + 2^14 + 2^13 = 2^15 2^14 + 2^13 + 2^13 = 8^5 2^14 + 2^13 + 2^13 = 2^15 2^14 + 2^14 + 8^5 = 4^8 2^14 + 2^14 + 8^5 = 2^16 2^14 + 2^14 + 2^15 = 4^8 2^14 + 2^14 + 2^15 = 2^16 2^14 + 2^15 + 4^7 = 4^8 2^14 + 2^15 + 4^7 = 2^16 2^14 + 2^15 + 2^14 = 4^8 2^14 + 2^15 + 2^14 = 2^16 2^15 + 2^14 + 4^7 = 4^8 2^15 + 2^14 + 4^7 = 2^16 2^15 + 2^14 + 2^14 = 4^8 2^15 + 2^14 + 2^14 = 2^16 2^15 + 2^15 + 4^8 = 2^17 2^15 + 2^15 + 2^16 = 2^17 2^15 + 2^16 + 8^5 = 2^17 2^15 + 2^16 + 2^15 = 2^17 2^16 + 2^15 + 8^5 = 2^17 2^16 + 2^15 + 2^15 = 2^17 2^16 + 2^16 + 2^17 = 8^6 2^16 + 2^16 + 2^17 = 4^9 2^16 + 2^16 + 2^17 = 2^18 2^16 + 2^17 + 4^8 = 8^6 2^16 + 2^17 + 4^8 = 4^9 2^16 + 2^17 + 4^8 = 2^18 2^16 + 2^17 + 2^16 = 8^6 2^16 + 2^17 + 2^16 = 4^9 2^16 + 2^17 + 2^16 = 2^18 2^17 + 2^16 + 4^8 = 8^6 2^17 + 2^16 + 4^8 = 4^9 2^17 + 2^16 + 4^8 = 2^18 2^17 + 2^16 + 2^16 = 8^6 2^17 + 2^16 + 2^16 = 4^9 2^17 + 2^16 + 2^16 = 2^18 2^17 + 2^17 + 8^6 = 2^19 2^17 + 2^17 + 4^9 = 2^19 2^17 + 2^17 + 2^18 = 2^19 2^17 + 2^18 + 2^17 = 2^19 2^18 + 2^17 + 2^17 = 2^19 2^18 + 2^18 + 2^19 = 16^5 2^18 + 2^18 + 2^19 = 4^10 2^18 + 2^18 + 2^19 = 2^20 2^18 + 2^19 + 8^6 = 16^5 2^18 + 2^19 + 8^6 = 4^10 2^18 + 2^19 + 8^6 = 2^20 2^18 + 2^19 + 4^9 = 16^5 2^18 + 2^19 + 4^9 = 4^10 2^18 + 2^19 + 4^9 = 2^20 2^18 + 2^19 + 2^18 = 16^5 2^18 + 2^19 + 2^18 = 4^10 2^18 + 2^19 + 2^18 = 2^20 2^19 + 2^18 + 8^6 = 16^5 2^19 + 2^18 + 8^6 = 4^10 2^19 + 2^18 + 8^6 = 2^20 2^19 + 2^18 + 4^9 = 16^5 2^19 + 2^18 + 4^9 = 4^10 2^19 + 2^18 + 4^9 = 2^20 2^19 + 2^18 + 2^18 = 16^5 2^19 + 2^18 + 2^18 = 4^10 2^19 + 2^18 + 2^18 = 2^20 2^19 + 2^19 + 16^5 = 8^7 2^19 + 2^19 + 4^10 = 8^7 2^19 + 2^19 + 2^20 = 8^7 2^19 + 2^20 + 2^19 = 8^7 2^20 + 2^19 + 2^19 = 8^7 2^20 + 2^20 + 8^7 = 4^11 2^9 + 4^5 + 2^9 = 2^11 2^10 + 4^5 + 2^11 = 4^6 2^10 + 4^5 + 2^11 = 2^12 2^11 + 4^5 + 4^5 = 4^6 2^11 + 4^5 + 4^5 = 2^12 2^11 + 4^5 + 2^10 = 4^6 2^11 + 4^5 + 2^10 = 2^12 2^11 + 4^6 + 2^11 = 2^13 2^12 + 4^6 + 2^13 = 4^7 2^12 + 4^6 + 2^13 = 2^14 2^13 + 4^6 + 4^6 = 4^7 2^13 + 4^6 + 4^6 = 2^14 2^13 + 4^6 + 2^12 = 4^7 2^13 + 4^6 + 2^12 = 2^14 2^13 + 4^7 + 2^13 = 8^5 2^13 + 4^7 + 2^13 = 2^15 2^14 + 4^7 + 8^5 = 4^8 2^14 + 4^7 + 8^5 = 2^16 2^14 + 4^7 + 2^15 = 4^8 2^14 + 4^7 + 2^15 = 2^16 2^15 + 4^7 + 4^7 = 4^8 2^15 + 4^7 + 4^7 = 2^16 2^15 + 4^7 + 2^14 = 4^8 2^15 + 4^7 + 2^14 = 2^16 2^15 + 4^8 + 8^5 = 2^17 2^15 + 4^8 + 2^15 = 2^17 2^16 + 4^8 + 2^17 = 8^6 2^16 + 4^8 + 2^17 = 4^9 2^16 + 4^8 + 2^17 = 2^18 2^17 + 4^8 + 4^8 = 8^6 2^17 + 4^8 + 4^8 = 4^9 2^17 + 4^8 + 4^8 = 2^18 2^17 + 4^8 + 2^16 = 8^6 2^17 + 4^8 + 2^16 = 4^9 2^17 + 4^8 + 2^16 = 2^18 2^17 + 4^9 + 2^17 = 2^19 2^18 + 4^9 + 2^19 = 16^5 2^18 + 4^9 + 2^19 = 4^10 2^18 + 4^9 + 2^19 = 2^20 2^19 + 4^9 + 8^6 = 16^5 2^19 + 4^9 + 8^6 = 4^10 2^19 + 4^9 + 8^6 = 2^20 2^19 + 4^9 + 4^9 = 16^5 2^19 + 4^9 + 4^9 = 4^10 2^19 + 4^9 + 4^9 = 2^20 2^19 + 4^9 + 2^18 = 16^5 2^19 + 4^9 + 2^18 = 4^10 2^19 + 4^9 + 2^18 = 2^20 2^19 + 4^10 + 2^19 = 8^7 2^20 + 4^10 + 8^7 = 4^11 2^14 + 8^5 + 4^7 = 4^8 2^14 + 8^5 + 4^7 = 2^16 2^14 + 8^5 + 2^14 = 4^8 2^14 + 8^5 + 2^14 = 2^16 2^15 + 8^5 + 4^8 = 2^17 2^15 + 8^5 + 2^16 = 2^17 2^16 + 8^5 + 8^5 = 2^17 2^16 + 8^5 + 2^15 = 2^17 2^17 + 8^6 + 2^17 = 2^19 2^18 + 8^6 + 2^19 = 16^5 2^18 + 8^6 + 2^19 = 4^10 2^18 + 8^6 + 2^19 = 2^20 2^19 + 8^6 + 8^6 = 16^5 2^19 + 8^6 + 8^6 = 4^10 2^19 + 8^6 + 8^6 = 2^20 2^19 + 8^6 + 4^9 = 16^5 2^19 + 8^6 + 4^9 = 4^10 2^19 + 8^6 + 4^9 = 2^20 2^19 + 8^6 + 2^18 = 16^5 2^19 + 8^6 + 2^18 = 4^10 2^19 + 8^6 + 2^18 = 2^20 2^20 + 8^7 + 16^5 = 4^11 2^20 + 8^7 + 4^10 = 4^11 2^20 + 8^7 + 2^20 = 4^11 2^19 + 16^5 + 2^19 = 8^7 2^20 + 16^5 + 8^7 = 4^11 241^5 + 241^5 + 241^6 = 723^5 241^5 + 241^6 + 241^5 = 723^5 241^6 + 241^5 + 241^5 = 723^5 241^10 + 241^10 + 241^11 = 174243^5 241^10 + 241^11 + 241^10 = 174243^5 241^11 + 241^10 + 241^10 = 174243^5 241^15 + 241^15 + 241^16 = 41992563^5 241^15 + 241^16 + 241^15 = 41992563^5 241^16 + 241^15 + 241^15 = 41992563^5 242^5 + 242^8 + 726^6 = 6534^5 242^6 + 242^7 + 726^5 = 2178^5 242^7 + 242^6 + 726^5 = 2178^5 242^8 + 242^5 + 726^6 = 6534^5 242^5 + 726^6 + 242^8 = 6534^5 242^6 + 726^5 + 242^7 = 2178^5 242^7 + 726^5 + 242^6 = 2178^5 242^8 + 726^6 + 242^5 = 6534^5 243^16 + 81^20 + 243^16 = 19683^9 243^16 + 81^20 + 81^20 = 19683^9 243^5 + 243^5 + 243^5 = 9^13 243^7 + 243^7 + 243^7 = 729^6 243^7 + 243^7 + 243^7 = 81^9 243^7 + 243^7 + 243^7 = 27^12 243^7 + 243^7 + 243^7 = 9^18 243^10 + 243^10 + 243^10 = 27^17 243^11 + 243^11 + 243^11 = 6561^7 243^11 + 243^11 + 243^11 = 2187^8 243^11 + 243^11 + 243^11 = 81^14 243^13 + 243^13 + 243^13 = 177147^6 243^13 + 243^13 + 243^13 = 729^11 243^15 + 243^15 + 243^15 = 81^19 243^16 + 243^16 + 243^16 = 19683^9 243^16 + 243^16 + 81^20 = 19683^9 243^18 + 243^18 + 729^15 = 1594323^7 243^18 + 243^18 + 729^15 = 2187^13 243^18 + 243^18 + 243^18 = 1594323^7 243^18 + 243^18 + 243^18 = 2187^13 243^19 + 243^19 + 243^19 = 43046721^6 243^19 + 243^19 + 243^19 = 531441^8 243^19 + 243^19 + 243^19 = 6561^12 243^19 + 243^19 + 243^19 = 729^16 243^18 + 729^15 + 729^15 = 1594323^7 243^18 + 729^15 + 729^15 = 2187^13 243^18 + 729^15 + 243^18 = 1594323^7 243^18 + 729^15 + 243^18 = 2187^13 244^6 + 61^8 + 122^7 = 305^6 244^6 + 122^7 + 61^8 = 305^6 245^5 + 245^5 + 735^5 = 245^6 245^5 + 735^5 + 245^5 = 245^6 25^8 + 10^15 + 250^5 = 75^8 25^8 + 250^5 + 1000^5 = 75^8 25^8 + 250^5 + 10^15 = 75^8 25^8 + 1000^5 + 250^5 = 75^8 250^5 + 5^16 + 1000^5 = 75^8 250^5 + 5^16 + 10^15 = 75^8 250^5 + 10^15 + 25^8 = 75^8 250^5 + 10^15 + 5^16 = 75^8 250^5 + 25^8 + 1000^5 = 75^8 250^5 + 25^8 + 10^15 = 75^8 250^5 + 1000^5 + 25^8 = 75^8 250^5 + 1000^5 + 5^16 = 75^8 254^7 + 127^8 + 127^9 = 508^7 254^7 + 127^9 + 127^8 = 508^7 254^8 + 254^8 + 254^9 = 508^8 254^8 + 254^9 + 254^8 = 508^8 254^9 + 254^8 + 254^8 = 508^8 254^16 + 254^16 + 254^17 = 129032^8 254^16 + 254^17 + 254^16 = 129032^8 254^17 + 254^16 + 254^16 = 129032^8 255^9 + 255^10 + 510^8 = 1020^8 255^10 + 255^9 + 510^8 = 1020^8 255^9 + 510^8 + 255^10 = 1020^8 255^10 + 510^8 + 255^9 = 1020^8 256^6 + 128^7 + 256^6 = 1024^5 256^6 + 128^7 + 256^6 = 32^10 256^6 + 128^7 + 64^8 = 1024^5 256^6 + 128^7 + 64^8 = 32^10 256^6 + 128^7 + 16^12 = 1024^5 256^6 + 128^7 + 16^12 = 32^10 256^6 + 128^7 + 8^16 = 1024^5 256^6 + 128^7 + 8^16 = 32^10 256^8 + 128^9 + 512^7 = 8192^5 256^8 + 128^9 + 512^7 = 32^13 256^8 + 128^9 + 128^9 = 8192^5 256^8 + 128^9 + 128^9 = 32^13 256^15 + 128^17 + 128^17 = 2048^11 256^6 + 256^6 + 128^7 = 1024^5 256^6 + 256^6 + 128^7 = 32^10 256^8 + 256^8 + 32^13 = 2048^6 256^8 + 256^8 + 32^13 = 64^11 256^19 + 256^19 + 512^17 = 4194304^7 256^19 + 256^19 + 512^17 = 16384^11 256^19 + 256^19 + 512^17 = 2048^14 256^8 + 512^7 + 512^7 = 8192^5 256^8 + 512^7 + 512^7 = 32^13 256^8 + 512^7 + 128^9 = 8192^5 256^8 + 512^7 + 128^9 = 32^13 256^19 + 512^17 + 256^19 = 4194304^7 256^19 + 512^17 + 256^19 = 16384^11 256^19 + 512^17 + 256^19 = 2048^14 257^7 + 514^7 + 514^7 = 257^8 258^6 + 129^6 + 258^6 = 129^7 258^7 + 129^8 + 516^7 = 129^9 258^6 + 258^6 + 129^6 = 129^7 258^8 + 258^8 + 516^8 = 258^9 258^7 + 516^7 + 129^8 = 129^9 258^8 + 516^8 + 258^8 = 258^9 260^6 + 65^7 + 130^6 = 65^8 260^6 + 130^6 + 65^7 = 65^8 260^7 + 130^7 + 130^7 = 130^8 27^5 + 3^15 + 27^5 = 9^8 27^5 + 3^15 + 27^5 = 3^16 27^5 + 3^15 + 3^15 = 9^8 27^5 + 3^15 + 3^15 = 3^16 27^6 + 3^18 + 27^6 = 3^19 27^6 + 3^18 + 9^9 = 3^19 27^6 + 3^18 + 3^18 = 3^19 27^6 + 9^9 + 27^6 = 3^19 27^6 + 9^9 + 9^9 = 3^19 27^6 + 9^9 + 3^18 = 3^19 27^8 + 9^12 + 81^6 = 243^5 27^8 + 9^12 + 27^8 = 243^5 27^8 + 9^12 + 9^12 = 243^5 27^5 + 27^5 + 27^5 = 9^8 27^5 + 27^5 + 27^5 = 3^16 27^5 + 27^5 + 3^15 = 9^8 27^5 + 27^5 + 3^15 = 3^16 27^6 + 27^6 + 27^6 = 3^19 27^6 + 27^6 + 9^9 = 3^19 27^6 + 27^6 + 3^18 = 3^19 27^7 + 27^7 + 27^7 = 9^11 27^8 + 27^8 + 81^6 = 243^5 27^8 + 27^8 + 27^8 = 243^5 27^8 + 27^8 + 9^12 = 243^5 27^9 + 27^9 + 27^9 = 81^7 27^9 + 27^9 + 27^9 = 9^14 27^11 + 27^11 + 27^11 = 9^17 27^13 + 27^13 + 27^13 = 6561^5 27^13 + 27^13 + 27^13 = 243^8 27^13 + 27^13 + 27^13 = 81^10 27^13 + 27^13 + 27^13 = 9^20 27^16 + 27^16 + 729^8 = 2187^7 27^16 + 27^16 + 81^12 = 2187^7 27^16 + 27^16 + 27^16 = 2187^7 27^17 + 27^17 + 27^17 = 81^13 27^18 + 27^18 + 729^9 = 177147^5 27^18 + 27^18 + 729^9 = 243^11 27^18 + 27^18 + 27^18 = 177147^5 27^18 + 27^18 + 27^18 = 243^11 27^8 + 81^6 + 81^6 = 243^5 27^8 + 81^6 + 27^8 = 243^5 27^8 + 81^6 + 9^12 = 243^5 27^16 + 81^12 + 729^8 = 2187^7 27^16 + 81^12 + 81^12 = 2187^7 27^16 + 81^12 + 27^16 = 2187^7 27^16 + 729^8 + 729^8 = 2187^7 27^16 + 729^8 + 81^12 = 2187^7 27^16 + 729^8 + 27^16 = 2187^7 27^18 + 729^9 + 729^9 = 177147^5 27^18 + 729^9 + 729^9 = 243^11 27^18 + 729^9 + 27^18 = 177147^5 27^18 + 729^9 + 27^18 = 243^11 270^6 + 270^7 + 810^6 = 90^9 270^7 + 270^6 + 810^6 = 90^9 270^6 + 810^6 + 270^7 = 90^9 270^7 + 810^6 + 270^6 = 90^9 276^5 + 552^5 + 828^5 = 276^6 276^5 + 828^5 + 552^5 = 276^6 288^7 + 288^7 + 288^7 = 3456^5 288^12 + 288^12 + 288^12 = 995328^5 288^17 + 288^17 + 288^17 = 286654464^5 3^5 + 3^5 + 3^5 = 3^6 3^6 + 3^6 + 3^6 = 3^7 3^7 + 3^7 + 3^7 = 3^8 3^8 + 3^8 + 3^8 = 3^9 3^9 + 3^9 + 3^9 = 9^5 3^9 + 3^9 + 3^9 = 3^10 3^10 + 3^10 + 9^5 = 3^11 3^10 + 3^10 + 3^10 = 3^11 3^11 + 3^11 + 3^11 = 9^6 3^11 + 3^11 + 3^11 = 3^12 3^12 + 3^12 + 9^6 = 3^13 3^12 + 3^12 + 3^12 = 3^13 3^13 + 3^13 + 3^13 = 9^7 3^13 + 3^13 + 3^13 = 3^14 3^14 + 3^14 + 9^7 = 27^5 3^14 + 3^14 + 9^7 = 3^15 3^14 + 3^14 + 3^14 = 27^5 3^14 + 3^14 + 3^14 = 3^15 3^15 + 3^15 + 27^5 = 9^8 3^15 + 3^15 + 27^5 = 3^16 3^15 + 3^15 + 3^15 = 9^8 3^15 + 3^15 + 3^15 = 3^16 3^16 + 3^16 + 9^8 = 3^17 3^16 + 3^16 + 3^16 = 3^17 3^17 + 3^17 + 3^17 = 27^6 3^17 + 3^17 + 3^17 = 9^9 3^17 + 3^17 + 3^17 = 3^18 3^18 + 3^18 + 27^6 = 3^19 3^18 + 3^18 + 9^9 = 3^19 3^18 + 3^18 + 3^18 = 3^19 3^19 + 3^19 + 3^19 = 81^5 3^19 + 3^19 + 3^19 = 9^10 3^19 + 3^19 + 3^19 = 3^20 3^20 + 3^20 + 81^5 = 27^7 3^20 + 3^20 + 9^10 = 27^7 3^20 + 3^20 + 3^20 = 27^7 3^10 + 9^5 + 9^5 = 3^11 3^10 + 9^5 + 3^10 = 3^11 3^12 + 9^6 + 9^6 = 3^13 3^12 + 9^6 + 3^12 = 3^13 3^14 + 9^7 + 9^7 = 27^5 3^14 + 9^7 + 9^7 = 3^15 3^14 + 9^7 + 3^14 = 27^5 3^14 + 9^7 + 3^14 = 3^15 3^16 + 9^8 + 9^8 = 3^17 3^16 + 9^8 + 3^16 = 3^17 3^18 + 9^9 + 27^6 = 3^19 3^18 + 9^9 + 9^9 = 3^19 3^18 + 9^9 + 3^18 = 3^19 3^20 + 9^10 + 81^5 = 27^7 3^20 + 9^10 + 9^10 = 27^7 3^20 + 9^10 + 3^20 = 27^7 3^15 + 27^5 + 27^5 = 9^8 3^15 + 27^5 + 27^5 = 3^16 3^15 + 27^5 + 3^15 = 9^8 3^15 + 27^5 + 3^15 = 3^16 3^18 + 27^6 + 27^6 = 3^19 3^18 + 27^6 + 9^9 = 3^19 3^18 + 27^6 + 3^18 = 3^19 3^20 + 81^5 + 81^5 = 27^7 3^20 + 81^5 + 9^10 = 27^7 3^20 + 81^5 + 3^20 = 27^7 30^5 + 30^5 + 30^6 = 60^5 30^5 + 30^6 + 30^5 = 60^5 30^6 + 30^5 + 30^5 = 60^5 30^10 + 30^10 + 30^11 = 1800^5 30^10 + 30^11 + 900^5 = 1800^5 30^10 + 30^11 + 30^10 = 1800^5 30^11 + 30^10 + 900^5 = 1800^5 30^11 + 30^10 + 30^10 = 1800^5 30^15 + 30^15 + 900^8 = 54000^5 30^15 + 30^15 + 30^16 = 54000^5 30^15 + 30^16 + 30^15 = 54000^5 30^16 + 30^15 + 30^15 = 54000^5 30^10 + 900^5 + 30^11 = 1800^5 30^11 + 900^5 + 900^5 = 1800^5 30^11 + 900^5 + 30^10 = 1800^5 30^15 + 900^8 + 30^15 = 54000^5 304^7 + 152^8 + 608^6 = 912^6 304^7 + 608^6 + 152^8 = 912^6 31^6 + 31^7 + 62^5 = 124^5 31^7 + 31^6 + 62^5 = 124^5 31^6 + 62^5 + 31^7 = 124^5 31^7 + 62^5 + 31^6 = 124^5 32^5 + 4^12 + 16^6 = 4^13 32^5 + 4^12 + 8^8 = 4^13 32^5 + 4^12 + 4^12 = 4^13 32^5 + 4^13 + 32^5 = 8^9 32^7 + 4^17 + 4^17 = 64^6 32^7 + 4^17 + 4^17 = 16^9 32^7 + 4^17 + 4^17 = 8^12 32^7 + 4^17 + 4^17 = 4^18 32^5 + 8^8 + 16^6 = 4^13 32^5 + 8^8 + 8^8 = 4^13 32^5 + 8^8 + 4^12 = 4^13 32^10 + 8^17 + 32^10 = 16^13 32^11 + 8^18 + 512^6 = 256^7 32^11 + 8^18 + 512^6 = 128^8 32^11 + 8^18 + 512^6 = 16^14 32^11 + 8^18 + 64^9 = 256^7 32^11 + 8^18 + 64^9 = 128^8 32^11 + 8^18 + 64^9 = 16^14 32^11 + 8^18 + 8^18 = 256^7 32^11 + 8^18 + 8^18 = 128^8 32^11 + 8^18 + 8^18 = 16^14 32^5 + 16^6 + 16^6 = 4^13 32^5 + 16^6 + 8^8 = 4^13 32^5 + 16^6 + 4^12 = 4^13 32^11 + 16^14 + 32^11 = 8^19 32^13 + 16^16 + 256^8 = 2048^6 32^13 + 16^16 + 256^8 = 64^11 32^13 + 16^16 + 16^16 = 2048^6 32^13 + 16^16 + 16^16 = 64^11 32^15 + 16^19 + 32^15 = 2048^7 32^15 + 16^19 + 32^15 = 128^11 32^5 + 32^5 + 4^13 = 8^9 32^10 + 32^10 + 8^17 = 16^13 32^11 + 32^11 + 256^7 = 8^19 32^11 + 32^11 + 128^8 = 8^19 32^11 + 32^11 + 16^14 = 8^19 32^15 + 32^15 + 16^19 = 2048^7 32^15 + 32^15 + 16^19 = 128^11 32^11 + 64^9 + 512^6 = 256^7 32^11 + 64^9 + 512^6 = 128^8 32^11 + 64^9 + 512^6 = 16^14 32^11 + 64^9 + 64^9 = 256^7 32^11 + 64^9 + 64^9 = 128^8 32^11 + 64^9 + 64^9 = 16^14 32^11 + 64^9 + 8^18 = 256^7 32^11 + 64^9 + 8^18 = 128^8 32^11 + 64^9 + 8^18 = 16^14 32^10 + 128^7 + 128^7 = 8^17 32^11 + 128^8 + 32^11 = 8^19 32^11 + 256^7 + 32^11 = 8^19 32^13 + 256^8 + 256^8 = 2048^6 32^13 + 256^8 + 256^8 = 64^11 32^13 + 256^8 + 16^16 = 2048^6 32^13 + 256^8 + 16^16 = 64^11 32^11 + 512^6 + 512^6 = 256^7 32^11 + 512^6 + 512^6 = 128^8 32^11 + 512^6 + 512^6 = 16^14 32^11 + 512^6 + 64^9 = 256^7 32^11 + 512^6 + 64^9 = 128^8 32^11 + 512^6 + 64^9 = 16^14 32^11 + 512^6 + 8^18 = 256^7 32^11 + 512^6 + 8^18 = 128^8 32^11 + 512^6 + 8^18 = 16^14 324^6 + 18^12 + 108^8 = 18^13 324^6 + 108^8 + 324^6 = 18^13 324^6 + 108^8 + 18^12 = 18^13 324^6 + 324^6 + 108^8 = 18^13 33^6 + 66^5 + 132^5 = 33^7 33^6 + 132^5 + 66^5 = 33^7 352^12 + 352^12 + 704^10 = 1486848^5 352^12 + 704^10 + 352^12 = 1486848^5 358^5 + 179^6 + 358^5 = 537^5 358^5 + 358^5 + 179^6 = 537^5 36^9 + 72^9 + 216^7 = 648^6 36^9 + 216^7 + 72^9 = 648^6 364^6 + 364^7 + 364^7 = 1092^6 364^7 + 364^6 + 364^7 = 1092^6 364^7 + 364^7 + 364^6 = 1092^6 364^12 + 364^13 + 364^13 = 397488^6 364^13 + 364^12 + 364^13 = 397488^6 364^13 + 364^13 + 364^12 = 397488^6 364^18 + 364^19 + 364^19 = 144685632^6 364^19 + 364^18 + 364^19 = 144685632^6 364^19 + 364^19 + 364^18 = 144685632^6 4^5 + 2^9 + 2^9 = 2^11 4^5 + 2^10 + 2^11 = 4^6 4^5 + 2^10 + 2^11 = 2^12 4^5 + 2^11 + 4^5 = 4^6 4^5 + 2^11 + 4^5 = 2^12 4^5 + 2^11 + 2^10 = 4^6 4^5 + 2^11 + 2^10 = 2^12 4^6 + 2^11 + 2^11 = 2^13 4^6 + 2^12 + 2^13 = 4^7 4^6 + 2^12 + 2^13 = 2^14 4^6 + 2^13 + 4^6 = 4^7 4^6 + 2^13 + 4^6 = 2^14 4^6 + 2^13 + 2^12 = 4^7 4^6 + 2^13 + 2^12 = 2^14 4^7 + 2^13 + 2^13 = 8^5 4^7 + 2^13 + 2^13 = 2^15 4^7 + 2^14 + 8^5 = 4^8 4^7 + 2^14 + 8^5 = 2^16 4^7 + 2^14 + 2^15 = 4^8 4^7 + 2^14 + 2^15 = 2^16 4^7 + 2^15 + 4^7 = 4^8 4^7 + 2^15 + 4^7 = 2^16 4^7 + 2^15 + 2^14 = 4^8 4^7 + 2^15 + 2^14 = 2^16 4^8 + 2^15 + 8^5 = 2^17 4^8 + 2^15 + 2^15 = 2^17 4^8 + 2^16 + 2^17 = 8^6 4^8 + 2^16 + 2^17 = 4^9 4^8 + 2^16 + 2^17 = 2^18 4^8 + 2^17 + 4^8 = 8^6 4^8 + 2^17 + 4^8 = 4^9 4^8 + 2^17 + 4^8 = 2^18 4^8 + 2^17 + 2^16 = 8^6 4^8 + 2^17 + 2^16 = 4^9 4^8 + 2^17 + 2^16 = 2^18 4^9 + 2^17 + 2^17 = 2^19 4^9 + 2^18 + 2^19 = 16^5 4^9 + 2^18 + 2^19 = 4^10 4^9 + 2^18 + 2^19 = 2^20 4^9 + 2^19 + 8^6 = 16^5 4^9 + 2^19 + 8^6 = 4^10 4^9 + 2^19 + 8^6 = 2^20 4^9 + 2^19 + 4^9 = 16^5 4^9 + 2^19 + 4^9 = 4^10 4^9 + 2^19 + 4^9 = 2^20 4^9 + 2^19 + 2^18 = 16^5 4^9 + 2^19 + 2^18 = 4^10 4^9 + 2^19 + 2^18 = 2^20 4^10 + 2^19 + 2^19 = 8^7 4^10 + 2^20 + 8^7 = 4^11 4^5 + 4^5 + 2^11 = 4^6 4^5 + 4^5 + 2^11 = 2^12 4^6 + 4^6 + 2^13 = 4^7 4^6 + 4^6 + 2^13 = 2^14 4^7 + 4^7 + 8^5 = 4^8 4^7 + 4^7 + 8^5 = 2^16 4^7 + 4^7 + 2^15 = 4^8 4^7 + 4^7 + 2^15 = 2^16 4^8 + 4^8 + 2^17 = 8^6 4^8 + 4^8 + 2^17 = 4^9 4^8 + 4^8 + 2^17 = 2^18 4^9 + 4^9 + 2^19 = 16^5 4^9 + 4^9 + 2^19 = 4^10 4^9 + 4^9 + 2^19 = 2^20 4^10 + 4^10 + 8^7 = 4^11 4^12 + 4^12 + 32^5 = 4^13 4^13 + 4^13 + 8^9 = 16^7 4^13 + 4^13 + 8^9 = 4^14 4^16 + 4^16 + 8^11 = 4^17 4^17 + 4^17 + 128^5 = 64^6 4^17 + 4^17 + 128^5 = 16^9 4^17 + 4^17 + 128^5 = 8^12 4^17 + 4^17 + 128^5 = 4^18 4^17 + 4^17 + 32^7 = 64^6 4^17 + 4^17 + 32^7 = 16^9 4^17 + 4^17 + 32^7 = 8^12 4^17 + 4^17 + 32^7 = 4^18 4^19 + 4^19 + 8^13 = 256^5 4^19 + 4^19 + 8^13 = 32^8 4^19 + 4^19 + 8^13 = 16^10 4^19 + 4^19 + 8^13 = 4^20 4^7 + 8^5 + 4^7 = 4^8 4^7 + 8^5 + 4^7 = 2^16 4^7 + 8^5 + 2^14 = 4^8 4^7 + 8^5 + 2^14 = 2^16 4^8 + 8^5 + 8^5 = 2^17 4^8 + 8^5 + 2^15 = 2^17 4^9 + 8^6 + 2^19 = 16^5 4^9 + 8^6 + 2^19 = 4^10 4^9 + 8^6 + 2^19 = 2^20 4^10 + 8^7 + 16^5 = 4^11 4^10 + 8^7 + 4^10 = 4^11 4^10 + 8^7 + 2^20 = 4^11 4^12 + 8^8 + 32^5 = 4^13 4^13 + 8^9 + 4^13 = 16^7 4^13 + 8^9 + 4^13 = 4^14 4^16 + 8^11 + 16^8 = 4^17 4^16 + 8^11 + 4^16 = 4^17 4^17 + 8^11 + 8^11 = 128^5 4^17 + 8^11 + 8^11 = 32^7 4^19 + 8^13 + 4^19 = 256^5 4^19 + 8^13 + 4^19 = 32^8 4^19 + 8^13 + 4^19 = 16^10 4^19 + 8^13 + 4^19 = 4^20 4^10 + 16^5 + 8^7 = 4^11 4^12 + 16^6 + 32^5 = 4^13 4^16 + 16^8 + 8^11 = 4^17 4^12 + 32^5 + 16^6 = 4^13 4^12 + 32^5 + 8^8 = 4^13 4^12 + 32^5 + 4^12 = 4^13 4^13 + 32^5 + 32^5 = 8^9 4^17 + 32^7 + 4^17 = 64^6 4^17 + 32^7 + 4^17 = 16^9 4^17 + 32^7 + 4^17 = 8^12 4^17 + 32^7 + 4^17 = 4^18 4^17 + 128^5 + 4^17 = 64^6 4^17 + 128^5 + 4^17 = 16^9 4^17 + 128^5 + 4^17 = 8^12 4^17 + 128^5 + 4^17 = 4^18 420^5 + 210^5 + 210^6 = 630^5 420^5 + 210^6 + 210^5 = 630^5 422^6 + 211^7 + 844^5 = 1899^5 422^6 + 844^5 + 211^7 = 1899^5 48^13 + 8^17 + 96^10 = 23552^5 48^12 + 96^9 + 192^9 = 13824^5 48^12 + 96^9 + 192^9 = 24^15 48^13 + 96^10 + 8^17 = 23552^5 48^12 + 192^9 + 96^9 = 13824^5 48^12 + 192^9 + 96^9 = 24^15 5^8 + 10^6 + 10^7 = 15^6 5^8 + 10^7 + 10^6 = 15^6 5^16 + 10^15 + 250^5 = 75^8 5^16 + 250^5 + 1000^5 = 75^8 5^16 + 250^5 + 10^15 = 75^8 5^16 + 1000^5 + 250^5 = 75^8 510^8 + 255^9 + 255^10 = 1020^8 510^8 + 255^10 + 255^9 = 1020^8 510^9 + 510^9 + 510^10 = 1020^9 510^9 + 510^10 + 510^9 = 1020^9 510^10 + 510^9 + 510^9 = 1020^9 510^18 + 510^18 + 510^19 = 520200^9 510^18 + 510^19 + 510^18 = 520200^9 510^19 + 510^18 + 510^18 = 520200^9 512^6 + 8^18 + 32^11 = 256^7 512^6 + 8^18 + 32^11 = 128^8 512^6 + 8^18 + 32^11 = 16^14 512^7 + 16^16 + 512^7 = 8192^5 512^7 + 16^16 + 512^7 = 32^13 512^7 + 16^16 + 128^9 = 8192^5 512^7 + 16^16 + 128^9 = 32^13 512^6 + 32^11 + 512^6 = 256^7 512^6 + 32^11 + 512^6 = 128^8 512^6 + 32^11 + 512^6 = 16^14 512^6 + 32^11 + 64^9 = 256^7 512^6 + 32^11 + 64^9 = 128^8 512^6 + 32^11 + 64^9 = 16^14 512^6 + 32^11 + 8^18 = 256^7 512^6 + 32^11 + 8^18 = 128^8 512^6 + 32^11 + 8^18 = 16^14 512^6 + 64^9 + 32^11 = 256^7 512^6 + 64^9 + 32^11 = 128^8 512^6 + 64^9 + 32^11 = 16^14 512^7 + 128^9 + 256^8 = 8192^5 512^7 + 128^9 + 256^8 = 32^13 512^7 + 128^9 + 16^16 = 8192^5 512^7 + 128^9 + 16^16 = 32^13 512^11 + 128^14 + 128^14 = 1048576^5 512^11 + 128^14 + 128^14 = 1024^10 512^11 + 128^14 + 128^14 = 32^20 512^7 + 256^8 + 512^7 = 8192^5 512^7 + 256^8 + 512^7 = 32^13 512^7 + 256^8 + 128^9 = 8192^5 512^7 + 256^8 + 128^9 = 32^13 512^17 + 256^19 + 256^19 = 4194304^7 512^17 + 256^19 + 256^19 = 16384^11 512^17 + 256^19 + 256^19 = 2048^14 512^6 + 512^6 + 32^11 = 256^7 512^6 + 512^6 + 32^11 = 128^8 512^6 + 512^6 + 32^11 = 16^14 512^7 + 512^7 + 256^8 = 8192^5 512^7 + 512^7 + 256^8 = 32^13 512^7 + 512^7 + 16^16 = 8192^5 512^7 + 512^7 + 16^16 = 32^13 514^7 + 257^7 + 514^7 = 257^8 514^7 + 514^7 + 257^7 = 257^8 516^7 + 129^8 + 258^7 = 129^9 516^7 + 258^7 + 129^8 = 129^9 516^8 + 258^8 + 258^8 = 258^9 552^5 + 276^5 + 828^5 = 276^6 552^5 + 828^5 + 276^5 = 276^6 6^18 + 72^9 + 216^7 = 648^6 6^18 + 216^7 + 72^9 = 648^6 608^6 + 152^8 + 304^7 = 912^6 608^6 + 304^7 + 152^8 = 912^6 61^8 + 122^7 + 244^6 = 305^6 61^8 + 244^6 + 122^7 = 305^6 614^5 + 614^5 + 921^5 = 307^6 614^5 + 921^5 + 614^5 = 307^6 62^5 + 31^6 + 31^7 = 124^5 62^5 + 31^7 + 31^6 = 124^5 62^6 + 62^6 + 62^7 = 124^6 62^6 + 62^7 + 62^6 = 124^6 62^7 + 62^6 + 62^6 = 124^6 62^12 + 62^12 + 62^13 = 7688^6 62^12 + 62^13 + 62^12 = 7688^6 62^13 + 62^12 + 62^12 = 7688^6 62^18 + 62^18 + 62^19 = 476656^6 62^18 + 62^19 + 62^18 = 476656^6 62^19 + 62^18 + 62^18 = 476656^6 63^7 + 63^8 + 126^6 = 252^6 63^8 + 63^7 + 126^6 = 252^6 63^7 + 126^6 + 63^8 = 252^6 63^8 + 126^6 + 63^7 = 252^6 64^8 + 8^16 + 128^7 = 1024^5 64^8 + 8^16 + 128^7 = 32^10 64^9 + 8^18 + 32^11 = 256^7 64^9 + 8^18 + 32^11 = 128^8 64^9 + 8^18 + 32^11 = 16^14 64^8 + 16^12 + 128^7 = 1024^5 64^8 + 16^12 + 128^7 = 32^10 64^9 + 32^11 + 512^6 = 256^7 64^9 + 32^11 + 512^6 = 128^8 64^9 + 32^11 + 512^6 = 16^14 64^9 + 32^11 + 64^9 = 256^7 64^9 + 32^11 + 64^9 = 128^8 64^9 + 32^11 + 64^9 = 16^14 64^9 + 32^11 + 8^18 = 256^7 64^9 + 32^11 + 8^18 = 128^8 64^9 + 32^11 + 8^18 = 16^14 64^8 + 64^8 + 128^7 = 1024^5 64^8 + 64^8 + 128^7 = 32^10 64^9 + 64^9 + 32^11 = 256^7 64^9 + 64^9 + 32^11 = 128^8 64^9 + 64^9 + 32^11 = 16^14 64^8 + 128^7 + 256^6 = 1024^5 64^8 + 128^7 + 256^6 = 32^10 64^8 + 128^7 + 64^8 = 1024^5 64^8 + 128^7 + 64^8 = 32^10 64^8 + 128^7 + 16^12 = 1024^5 64^8 + 128^7 + 16^12 = 32^10 64^8 + 128^7 + 8^16 = 1024^5 64^8 + 128^7 + 8^16 = 32^10 64^20 + 128^17 + 128^17 = 2048^11 64^8 + 256^6 + 128^7 = 1024^5 64^8 + 256^6 + 128^7 = 32^10 65^5 + 130^5 + 130^5 = 65^6 65^7 + 130^6 + 260^6 = 65^8 65^7 + 260^6 + 130^6 = 65^8 66^5 + 33^6 + 132^5 = 33^7 66^6 + 66^6 + 132^6 = 66^7 66^5 + 132^5 + 33^6 = 33^7 66^6 + 132^6 + 66^6 = 66^7 704^10 + 352^12 + 352^12 = 1486848^5 72^9 + 6^18 + 216^7 = 648^6 72^9 + 36^9 + 216^7 = 648^6 726^5 + 242^6 + 242^7 = 2178^5 726^5 + 242^7 + 242^6 = 2178^5 726^6 + 242^5 + 242^8 = 6534^5 726^6 + 242^8 + 242^5 = 6534^5 727^6 + 727^6 + 727^7 = 2181^6 727^6 + 727^7 + 727^6 = 2181^6 727^7 + 727^6 + 727^6 = 2181^6 727^12 + 727^12 + 727^13 = 1585587^6 727^12 + 727^13 + 727^12 = 1585587^6 727^13 + 727^12 + 727^12 = 1585587^6 727^18 + 727^18 + 727^19 = 1152721749^6 727^18 + 727^19 + 727^18 = 1152721749^6 727^19 + 727^18 + 727^18 = 1152721749^6 729^8 + 27^16 + 729^8 = 2187^7 729^8 + 27^16 + 81^12 = 2187^7 729^8 + 27^16 + 27^16 = 2187^7 729^9 + 27^18 + 729^9 = 177147^5 729^9 + 27^18 + 729^9 = 243^11 729^9 + 27^18 + 27^18 = 177147^5 729^9 + 27^18 + 27^18 = 243^11 729^8 + 81^12 + 729^8 = 2187^7 729^8 + 81^12 + 81^12 = 2187^7 729^8 + 81^12 + 27^16 = 2187^7 729^15 + 243^18 + 729^15 = 1594323^7 729^15 + 243^18 + 729^15 = 2187^13 729^15 + 243^18 + 243^18 = 1594323^7 729^15 + 243^18 + 243^18 = 2187^13 729^8 + 729^8 + 729^8 = 2187^7 729^8 + 729^8 + 81^12 = 2187^7 729^8 + 729^8 + 27^16 = 2187^7 729^9 + 729^9 + 729^9 = 177147^5 729^9 + 729^9 + 729^9 = 243^11 729^9 + 729^9 + 27^18 = 177147^5 729^9 + 729^9 + 27^18 = 243^11 729^14 + 729^14 + 729^14 = 129140163^5 729^14 + 729^14 + 729^14 = 243^17 729^15 + 729^15 + 729^15 = 1594323^7 729^15 + 729^15 + 729^15 = 2187^13 729^15 + 729^15 + 243^18 = 1594323^7 729^15 + 729^15 + 243^18 = 2187^13 729^19 + 729^19 + 729^19 = 94143178827^5 729^20 + 729^20 + 729^20 = 177147^11 735^5 + 245^5 + 245^5 = 245^6 768^7 + 768^7 + 768^7 = 384^8 768^15 + 768^15 + 768^15 = 294912^8 8^5 + 2^14 + 4^7 = 4^8 8^5 + 2^14 + 4^7 = 2^16 8^5 + 2^14 + 2^14 = 4^8 8^5 + 2^14 + 2^14 = 2^16 8^5 + 2^15 + 4^8 = 2^17 8^5 + 2^15 + 2^16 = 2^17 8^5 + 2^16 + 8^5 = 2^17 8^5 + 2^16 + 2^15 = 2^17 8^6 + 2^17 + 2^17 = 2^19 8^6 + 2^18 + 2^19 = 16^5 8^6 + 2^18 + 2^19 = 4^10 8^6 + 2^18 + 2^19 = 2^20 8^6 + 2^19 + 8^6 = 16^5 8^6 + 2^19 + 8^6 = 4^10 8^6 + 2^19 + 8^6 = 2^20 8^6 + 2^19 + 4^9 = 16^5 8^6 + 2^19 + 4^9 = 4^10 8^6 + 2^19 + 4^9 = 2^20 8^6 + 2^19 + 2^18 = 16^5 8^6 + 2^19 + 2^18 = 4^10 8^6 + 2^19 + 2^18 = 2^20 8^7 + 2^20 + 16^5 = 4^11 8^7 + 2^20 + 4^10 = 4^11 8^7 + 2^20 + 2^20 = 4^11 8^5 + 4^7 + 4^7 = 4^8 8^5 + 4^7 + 4^7 = 2^16 8^5 + 4^7 + 2^14 = 4^8 8^5 + 4^7 + 2^14 = 2^16 8^5 + 4^8 + 8^5 = 2^17 8^5 + 4^8 + 2^15 = 2^17 8^6 + 4^9 + 2^19 = 16^5 8^6 + 4^9 + 2^19 = 4^10 8^6 + 4^9 + 2^19 = 2^20 8^7 + 4^10 + 16^5 = 4^11 8^7 + 4^10 + 4^10 = 4^11 8^7 + 4^10 + 2^20 = 4^11 8^8 + 4^12 + 32^5 = 4^13 8^9 + 4^13 + 4^13 = 16^7 8^9 + 4^13 + 4^13 = 4^14 8^11 + 4^16 + 16^8 = 4^17 8^11 + 4^16 + 4^16 = 4^17 8^11 + 4^17 + 8^11 = 128^5 8^11 + 4^17 + 8^11 = 32^7 8^13 + 4^19 + 4^19 = 256^5 8^13 + 4^19 + 4^19 = 32^8 8^13 + 4^19 + 4^19 = 16^10 8^13 + 4^19 + 4^19 = 4^20 8^5 + 8^5 + 4^8 = 2^17 8^5 + 8^5 + 2^16 = 2^17 8^6 + 8^6 + 2^19 = 16^5 8^6 + 8^6 + 2^19 = 4^10 8^6 + 8^6 + 2^19 = 2^20 8^8 + 8^8 + 32^5 = 4^13 8^11 + 8^11 + 4^17 = 128^5 8^11 + 8^11 + 4^17 = 32^7 8^16 + 8^16 + 128^7 = 1024^5 8^16 + 8^16 + 128^7 = 32^10 8^18 + 8^18 + 32^11 = 256^7 8^18 + 8^18 + 32^11 = 128^8 8^18 + 8^18 + 32^11 = 16^14 8^7 + 16^5 + 16^5 = 4^11 8^7 + 16^5 + 4^10 = 4^11 8^7 + 16^5 + 2^20 = 4^11 8^8 + 16^6 + 32^5 = 4^13 8^11 + 16^8 + 16^8 = 4^17 8^11 + 16^8 + 4^16 = 4^17 8^16 + 16^12 + 128^7 = 1024^5 8^16 + 16^12 + 128^7 = 32^10 8^8 + 32^5 + 16^6 = 4^13 8^8 + 32^5 + 8^8 = 4^13 8^8 + 32^5 + 4^12 = 4^13 8^17 + 32^10 + 32^10 = 16^13 8^18 + 32^11 + 512^6 = 256^7 8^18 + 32^11 + 512^6 = 128^8 8^18 + 32^11 + 512^6 = 16^14 8^18 + 32^11 + 64^9 = 256^7 8^18 + 32^11 + 64^9 = 128^8 8^18 + 32^11 + 64^9 = 16^14 8^18 + 32^11 + 8^18 = 256^7 8^18 + 32^11 + 8^18 = 128^8 8^18 + 32^11 + 8^18 = 16^14 8^17 + 48^13 + 96^10 = 23552^5 8^16 + 64^8 + 128^7 = 1024^5 8^16 + 64^8 + 128^7 = 32^10 8^18 + 64^9 + 32^11 = 256^7 8^18 + 64^9 + 32^11 = 128^8 8^18 + 64^9 + 32^11 = 16^14 8^17 + 96^10 + 48^13 = 23552^5 8^16 + 128^7 + 256^6 = 1024^5 8^16 + 128^7 + 256^6 = 32^10 8^16 + 128^7 + 64^8 = 1024^5 8^16 + 128^7 + 64^8 = 32^10 8^16 + 128^7 + 16^12 = 1024^5 8^16 + 128^7 + 16^12 = 32^10 8^16 + 128^7 + 8^16 = 1024^5 8^16 + 128^7 + 8^16 = 32^10 8^16 + 256^6 + 128^7 = 1024^5 8^16 + 256^6 + 128^7 = 32^10 8^18 + 512^6 + 32^11 = 256^7 8^18 + 512^6 + 32^11 = 128^8 8^18 + 512^6 + 32^11 = 16^14 81^5 + 3^20 + 81^5 = 27^7 81^5 + 3^20 + 9^10 = 27^7 81^5 + 3^20 + 3^20 = 27^7 81^5 + 9^10 + 81^5 = 27^7 81^5 + 9^10 + 9^10 = 27^7 81^5 + 9^10 + 3^20 = 27^7 81^6 + 9^12 + 81^6 = 243^5 81^6 + 9^12 + 27^8 = 243^5 81^6 + 9^12 + 9^12 = 243^5 81^8 + 9^16 + 81^8 = 27^11 81^8 + 9^16 + 9^16 = 27^11 81^6 + 27^8 + 81^6 = 243^5 81^6 + 27^8 + 27^8 = 243^5 81^6 + 27^8 + 9^12 = 243^5 81^12 + 27^16 + 729^8 = 2187^7 81^12 + 27^16 + 81^12 = 2187^7 81^12 + 27^16 + 27^16 = 2187^7 81^5 + 81^5 + 81^5 = 27^7 81^5 + 81^5 + 9^10 = 27^7 81^5 + 81^5 + 3^20 = 27^7 81^6 + 81^6 + 81^6 = 243^5 81^6 + 81^6 + 27^8 = 243^5 81^6 + 81^6 + 9^12 = 243^5 81^8 + 81^8 + 81^8 = 27^11 81^8 + 81^8 + 9^16 = 27^11 81^11 + 81^11 + 81^11 = 19683^5 81^11 + 81^11 + 81^11 = 243^9 81^11 + 81^11 + 81^11 = 27^15 81^12 + 81^12 + 729^8 = 2187^7 81^12 + 81^12 + 81^12 = 2187^7 81^12 + 81^12 + 27^16 = 2187^7 81^14 + 81^14 + 81^14 = 27^19 81^16 + 81^16 + 81^16 = 1594323^5 81^16 + 81^16 + 81^16 = 243^13 81^19 + 81^19 + 81^19 = 177147^7 81^19 + 81^19 + 81^19 = 2187^11 81^20 + 81^20 + 243^16 = 19683^9 81^20 + 81^20 + 81^20 = 19683^9 81^20 + 243^16 + 243^16 = 19683^9 81^20 + 243^16 + 81^20 = 19683^9 81^12 + 729^8 + 729^8 = 2187^7 81^12 + 729^8 + 81^12 = 2187^7 81^12 + 729^8 + 27^16 = 2187^7 810^6 + 270^6 + 270^7 = 90^9 810^6 + 270^7 + 270^6 = 90^9 844^5 + 211^7 + 422^6 = 1899^5 844^5 + 422^6 + 211^7 = 1899^5 864^8 + 864^8 + 864^8 = 62208^5 864^13 + 864^13 + 864^13 = 53747712^5 864^18 + 864^18 + 864^18 = 46438023168^5 9^5 + 3^10 + 9^5 = 3^11 9^5 + 3^10 + 3^10 = 3^11 9^6 + 3^12 + 9^6 = 3^13 9^6 + 3^12 + 3^12 = 3^13 9^7 + 3^14 + 9^7 = 27^5 9^7 + 3^14 + 9^7 = 3^15 9^7 + 3^14 + 3^14 = 27^5 9^7 + 3^14 + 3^14 = 3^15 9^8 + 3^16 + 9^8 = 3^17 9^8 + 3^16 + 3^16 = 3^17 9^9 + 3^18 + 27^6 = 3^19 9^9 + 3^18 + 9^9 = 3^19 9^9 + 3^18 + 3^18 = 3^19 9^10 + 3^20 + 81^5 = 27^7 9^10 + 3^20 + 9^10 = 27^7 9^10 + 3^20 + 3^20 = 27^7 9^5 + 9^5 + 9^5 = 3^11 9^5 + 9^5 + 3^10 = 3^11 9^6 + 9^6 + 9^6 = 3^13 9^6 + 9^6 + 3^12 = 3^13 9^7 + 9^7 + 9^7 = 27^5 9^7 + 9^7 + 9^7 = 3^15 9^7 + 9^7 + 3^14 = 27^5 9^7 + 9^7 + 3^14 = 3^15 9^8 + 9^8 + 9^8 = 3^17 9^8 + 9^8 + 3^16 = 3^17 9^9 + 9^9 + 27^6 = 3^19 9^9 + 9^9 + 9^9 = 3^19 9^9 + 9^9 + 3^18 = 3^19 9^10 + 9^10 + 81^5 = 27^7 9^10 + 9^10 + 9^10 = 27^7 9^10 + 9^10 + 3^20 = 27^7 9^12 + 9^12 + 81^6 = 243^5 9^12 + 9^12 + 27^8 = 243^5 9^12 + 9^12 + 9^12 = 243^5 9^13 + 9^13 + 9^13 = 27^9 9^16 + 9^16 + 81^8 = 27^11 9^16 + 9^16 + 9^16 = 27^11 9^17 + 9^17 + 9^17 = 2187^5 9^17 + 9^17 + 9^17 = 243^7 9^19 + 9^19 + 9^19 = 27^13 9^9 + 27^6 + 27^6 = 3^19 9^9 + 27^6 + 9^9 = 3^19 9^9 + 27^6 + 3^18 = 3^19 9^12 + 27^8 + 81^6 = 243^5 9^12 + 27^8 + 27^8 = 243^5 9^12 + 27^8 + 9^12 = 243^5 9^10 + 81^5 + 81^5 = 27^7 9^10 + 81^5 + 9^10 = 27^7 9^10 + 81^5 + 3^20 = 27^7 9^12 + 81^6 + 81^6 = 243^5 9^12 + 81^6 + 27^8 = 243^5 9^12 + 81^6 + 9^12 = 243^5 9^16 + 81^8 + 81^8 = 27^11 9^16 + 81^8 + 9^16 = 27^11 921^5 + 614^5 + 614^5 = 307^6 992^5 + 496^6 + 496^6 = 1984^5 REFERENCES [1] http://www.bealconjecture.com [2] http://mathworld.wolfram.com/DiophantineEquation4thPowers.html