Landau prime ideal theorem: Difference between revisions

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A '''broken diagonal''' is a sequence of [[n]] diametrically-positioned cells in a [[panmagic square]] which spans the vertical boundary.  Examples of broken diagonals from the below [[Square (geometry)|square]] are as follows: 3,12,14,5; 10,1,7,16; 10,13,7,4; 15,8,2,9; 15,12,2,5; and 6,13,11,4.
 
<center>[[Image:PanmagicSquare-Order4.png]]</center>
 
Notice that because one of the properties of a panmagic square is that the broken diagonals add up to the same constant, the following pattern is evident:
 
'''<math>3+12+14+5=34</math>''';
 
'''<math>10+1+7+16=34</math>''';
 
'''<math>10+13+7+4=34</math>'''
 
One way to visualize a broken diagonal is to imagine a "ghost image" of the panmagic square adjacent to the original:
 
<center>[[Image:PanmagicSquare-Order4.png]][[Image:PanmagicSquare-Order4.png]]</center>
 
It is easy to see now how the set of numbers {3, 12, 14, 5} result to form a broken diagonal: once wrapped around the original square, it can now be seen starting with the first square of the ghost image and moving down to the left.
 
Although this specific article relates to broken diagonals in [[panmagic square]]s, the "broken diagonal" also has uses in [[Matrix (mathematics)|matrices]] and other areas of [[geometry]] such as determining the area of a [[polygon]] solely from its [[Cartesian coordinate system|Cartesian coordinates]].
 
 
{{numtheory-stub}}
[[Category:Magic squares]]

Latest revision as of 14:11, 25 November 2014

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