phil1882 Posted November 28, 2013 Report Share Posted November 28, 2013 whats the smallest number that can be expressed as the sum of 3 squares in 3 unique ways Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted December 2, 2013 Report Share Posted December 2, 2013 65 = 02+12+82 = 02+42+72 = 22+52+62 Quote Link to comment Share on other sites More sharing options...
0 Grimbal Posted November 29, 2013 Report Share Posted November 29, 2013 A bit cheating, but one candidate would be 125= 02 + 52 + 102 = 32 + 42 + 102 = 52 + 62 + 82 Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted November 29, 2013 Report Share Posted November 29, 2013 54 = 12+22+72 = 22+52+52 = 32+32+62 Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted November 29, 2013 Report Share Posted November 29, 2013 -125 = 02 + (sqrt(-25))2 + (sqrt(-100)2 = (sqrt(-9))2 + (sqrt(-16))2 + (sqrt(-100))2 = (sqrt(-25))2 + (sqrt(-36)2 + (sqrt(-64))2 so simply put any answer that can define a number using three squares has an equally negative solution. So whatever the biggest number we can make using three squares in three different ways naturally leads us to the smallest square number in three different ways. 1 Quote Link to comment Share on other sites More sharing options...
0 Grimbal Posted November 29, 2013 Report Share Posted November 29, 2013 The answer depends whether you accept 0 as a square and whether you accept the same number twice. 1 Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted November 29, 2013 Report Share Posted November 29, 2013 02 + 02 + 02 = (sqrt(-1)2 + (12) + 02 = 22+22+(sqrt(-8))2=0 1 Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted November 29, 2013 Report Share Posted November 29, 2013 101 = 12+62+82 = 22+42+92 = 42+62+72 Quote Link to comment Share on other sites More sharing options...
0 Perhaps check it again Posted December 2, 2013 Report Share Posted December 2, 2013 (edited) @ BMAD, your solutions do not count, because sqrt(-25), sqrt(-100), sqrt(-9), etc. are not integers. A square (number) is the square of an integer and is necessarily nonnegative. @ Grimbal, it is not a question of whether someone accepts 0 as a square (number). It is a fact. Square numbers are the squares of integers. The phrase "in 3 unique ways" is ambiguous. Edited December 2, 2013 by Perhaps check it again Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted December 2, 2013 Report Share Posted December 2, 2013 41 = 02+42+52 = 12+22+62 = 32+42+42 1 Quote Link to comment Share on other sites More sharing options...
0 BMAD Posted December 2, 2013 Report Share Posted December 2, 2013 @PerhapsCheckITAgain I see no requirement that the numbers be square numbers just that i square three numbers 1 Quote Link to comment Share on other sites More sharing options...
0 Prime Posted December 2, 2013 Report Share Posted December 2, 2013 Barring complex numbers and limiting ourselves to integers... 02+12+12=02+12+(-1)2=02+(-1)2+(-1)2=2 1 Quote Link to comment Share on other sites More sharing options...
0 phil1882 Posted December 2, 2013 Author Report Share Posted December 2, 2013 i like this anwer the best though i was aiming for non zero answers. 1 Quote Link to comment Share on other sites More sharing options...
0 Perhaps check it again Posted December 2, 2013 Report Share Posted December 2, 2013 (edited) @PerhapsCheckITAgain I see no requirement that the numbers be square numbers just that i square three numbers The problem stated that they are "squares." "Squares" in the integer sense *means* unambiguously that the squares of integers are the only ones permitted, that is, 0, 1, 4, 9, 16, ... Edited December 2, 2013 by Perhaps check it again Quote Link to comment Share on other sites More sharing options...
0 Perhaps check it again Posted December 2, 2013 Report Share Posted December 2, 2013 (edited) Barring complex numbers and limiting ourselves to integers... 02+12+12=02+12+(-1)2=02+(-1)2+(-1)2=2 User Prime, your solution doesn't count, because it amounts the *same* solution repeated, that is, 0^2 + 1^2 + 1^2 = 2, that is, 0 + 1 + 1 = 2. Your 2nd and 3rd expressions don't give different sums of squares than your 1st expression. Edited December 2, 2013 by Perhaps check it again Quote Link to comment Share on other sites More sharing options...
0 Perhaps check it again Posted December 2, 2013 Report Share Posted December 2, 2013 i like this anwer the best though i was aiming for non zero answers. phil1882, what Prime offered does not count as I explained. Quote Link to comment Share on other sites More sharing options...
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phil1882
whats the smallest number that can be expressed as the sum of 3 squares in 3 unique ways
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