Magic squares of 7th powers
See also the Magic
squares of cubes general page
Magic squares of 7th powers have been known since 20042005, using very big 65536 x 65536 heptamagic squares, directly raising their integers to the seventh power.
But is it possible to construct smaller squares of 7th powers? Hopefully, yes! In 2012 and 2013, using methods similar to those already used for squares of 6th powers, Jaroslaw Wroblewski and Toshihiro Shirakawa constructed these squares:
Square 
Order 
Date 
Author 
S7 
MaxNb 
Semimagic 
March 2012 
Jaroslaw Wroblewski 
3.54e+375 
(3.81e+53)^7 

April 2013 
Toshihiro Shirakawa 
1.97e+250 
(5.59e+35)^7 

May 2013 
3.03e+221 
(4.20e+31)^7 

Magic 
September 2013 
3.14e+51 
18421557^7 
The best known = the smallest known magic and semimagic are these two
squares below from
Toshihiro Shirakawa.
In his 144x144 magic square, amazing and fun: the 52
digits of S7
are exactly the 52 first digits of the famous number Pi...
YES,
!!!!
But not a coincidence... S7 = T1xT2... and in his construction method,
he chose T1 and T2 for that.
47^{7} 
1857420^{7} 
3467933^{7} 
3626244^{7} 
... 
26085^{7} 
510^{7} 
3674778^{7} 
3949113^{7} 
... 
63215^{7} 
283050^{7} 
1009^{7} 
4184658^{7} 
... 
67022^{7} 
685950^{7} 
559995^{7} 
1149^{7} 
... 
... 
... 
... 
... 
... 
36677008731553118198143136663400^{7} 
28445864857805223685028505333000^{7} 
22998784353119117021937940482000^{7} 
15614964113433505767526285906200^{7} 
... 
36129590690783668672797716713200^{7} 
28021299710673802435998229134000^{7} 
22655518915012861543998568236000^{7} 
15381904947561363890399027907600^{7} 
... 
33392500486936421046070616962200^{7} 
25898473975016696190846848139000^{7} 
20939191724481584154301707006000^{7} 
14216609118200654504762737914600^{7} 
... 
30381701262704448656670807236100^{7} 
23563365665793879321180329044500^{7} 
19051231814897179025635159653000^{7} 
12934783705903874180562818922300^{7} 
... 
... 
... 
... 
... 
... 
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