1.Introduction

It seems that W. Lenhart and A. B. Evans teated the problrem to find four positive numbers
such that the sum of any two is a cube.([1].Dickson)

Lenhart used following relation.

(95/7)^3 + (248/7)^3 = (149/12)^3 + (427/12)^3 = (341899/30291)^3 + (1081640/30291)^3 

They gave some large solutions (22 and 24 digits).  

I searched many numeric solutions. These are smaller than theirs.
Smallest solution,  [A, B, C, D]=[51114333631, 7071213244, 2510949380, 1493580092], was found.


2. Method



A + B = p^3................................................................(1)  
A + C = q^3................................................................(2)  
B + D = s^3................................................................(3)  
C + D = t^3................................................................(4)  
B + C = u^3................................................................(5)  
A + D = v^3................................................................(6)  
   
From (1),(2), and (5), A =(p^3 + q^3 - u^3)/2..............................(7)

Similarly, D =(s^3 + t^3 - u^3)/2..........................................(8)

From (1), (3), and B = p^3 - A = s^3 - D, then p^3 - s^3 = A - D...........(9)   

Similarly, C = q^3 - A = t^3 - D, then q^3 - t^3 = A - D..................(10)   

From (9), and (10), p^3 + t^3 = q^3 + s^3.................................(11)   

From (1)+(2)+(3)+(4)+(5)+(6), p^3 + q^3 + s^3 + t^3 = 2(u^3 + v^3)........(12)   

Hence, p^3 + t^3 = q^3 + s^3 = u^3 + v^3..................................(13)
                      
Accordingly, we must find the integer solutions of (13).

Many solutions were found by brute force search.

3. Result

Search method: Brute force.
{p, t, q, s, u, v}<100000
{A, B, C, D} is sorted in descending order.

[A, B, C, D]=[51114333631, 7071213244, 2510949380, 1493580092] is a smallest solution.

A+B=3875^3, A+C=3771^3, A+D=3747^3, B+C=2124^3, B+D=2046^3, C+D=1588^3.

One {p, t, q, s, u, v} sometimes leads to multiple solutions by interchanging six cubes as follows.
[66386  51262  65544  52620  62700  56544] leads to [163827996114820, 128741810605636, 117750072394364, 16955834122364] and
[196682022496228, 95887784224228, 84896046012956, 49809860503772].

[96282 41606 96852 38276 98175 26369] leads to  [109763784860183/2, 1782722433358567/2, 2388889316969/2, 34281116285849/2].
However, this result is not an integer,  multiplying by 8, we obtain [192564, 83212, 193704, 76552, 196350, 52738],
[7130889733434268, 439055139440732, 137124465143396, 9555557267876].


            {p, t, q, s, u, v}              [         A                 B                C                 D      ]  
[   3747   2124   3771   2046   3875   1588][      51114333631        7071213244       2510949380       1493580092]
[  18419  14917  18762  14364  19791  12201][    5518494262093     2233317468578    1085966964635     730327591966]
[  21978  13862  22592  12048  23096   9864][   10593628833748     1726359954988     937293264940      22459099604]
[  23328  12696  23706  11238  23792  10840][   12371699929684     1095982079404     950466050132     323294653868]
[  31286  17476  31350  17268  32768   9190][   30329251118828     4855120970004     482234256172     293917302828]
[  36945  35660  40457  30918  42009  27866][   47503935017361    26631703189368   18714822306632    2923515591264]
[  36350  19378  36420  19128  37860  10128][   47649632272924     6618119383076     658453015076     380440602076]
[  42900  38688  45422  35074  46996  32096][   69801278935356    33995215320580   23911487556092    9152310064644]
[  55620  39575  56162  38463  57495  35300][  152611348285764    37448437276611   24533160957764   19453816042236]
[  62700  56544  65544  52620  66386  51262][  163827996114820   128741810605636  117750072394364   16955834122364]
[  62700  56544  65544  52620  66386  51262][  196682022496228    95887784224228   84896046012956   49809860503772]
[  68962  65163  72355  60900  81310  40635][  319833063972064   217733048118936   58963162166811    8133480881064]
[  71434  68849  72105  68112  86049  37734][  323756295279385   313387536296264   51127047278240    2600688084664]
[  71032  53820  72954  50134  77532  36400][  349224335725716   116836877387052   39057725580948    9170818419052]
[  73890  71320  80914  61836  84018  55732][  359707779688388   233377325965444  170042278903556    3064424279612]
[  79662  64338  84540  55140  84800  54520][  473844617600764   135955574399236  130363743063236   31693162344764]
[  86572  60678  88470  56440  91158  48592][  613273509876780   144229503239532   79175951546220   35558626744468]
[  88596  73968  89194  73094 100116  45888][  654187292589524   349296745771372   55401785523860   41224968143212]
[  96828  51047  97140  49895  99495  38240][  884269392987776   100656985724599   32361093356224   23557166867776]
[ 109328  89848 113710  82514 117368  74608][ 1180863411451420   435909828208612  289402806359580  125891707804132]
[ 109180  61010 111650  51520 111762  50988][ 1280347935863364   115642667039364  111449981261636   21107404768636]
[ 113058  87800 115746  82976 122792  64674][ 1362635635299012   488806824118076  188029327881924   82484311292100]
[ 126106 121046 138196 104456 143392  94004][ 1907012281820244  1041308724272044  732273501933292   98416534578772]
[ 126960  71700 127262  70738 129460  62600][ 1931110159442364   238625407093636  129976405906364  115337970093636]
[ 127650 117070 132280 111060 149776  68724][ 2035024193034788  1324878374925788  279611035317212   44971604090212]
[ 142868 137698 144210 136224 172098  75468][ 2742680763076900  2354469889528292  256385977384100  173435905519132]
[ 152858  97126 156896  85528 160080  72792][ 3524060226903380   578086845608620  338147378307756   47553791509332]
[ 152988  84912 154864  78236 158234  61366][ 3531862259621460   430000435535444  182219138503084   48872082472812]
[ 156038 109962 159800 101580 169160  66060][ 3795784689415436  1044746573880564  284874502584564    3406290431436]
[ 176740 136820 178250 134230 195796  83204][ 5304183604887668  2201866264830668  359364785737332  216648653136332]
[ 189864  85590 190566  81972 194006  55312][ 6797769916157356   504291552794860  122710668084140   46511824047188]
[ 192564  83212 193704  76552 196350  52738][ 7130889733434268   439055139440732  137124465143396    9555557267876]






    
    
4. Reference

[1].L.E. Dickson:History of theory of numbers, vol 2.
    
    
    
    


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