1) Prove that an infinite number of AP's exist, such that each of the terms is a perfect square.
2) How many ways are there to place the 1st 64 nos. on the 64 squares of a std. chessboard, such that the terms in each col. and in each row are in AP ? (My ISI interview Qsn. Luckily I cud do it.)
1There is no way to arrange them on a chess board in the above mentioned constraint.
If you start with a even no. on the first cell, then if difference is even, you get only even nos. starting with the least 2,4,6,8,10,12,14,16. then the corresponding column has to have difference 16 that gets you 2,18,34,50,66 not possible(as odd difference yield a repeat of even no in 2-16)
341Are you sure about qn 1 because the book Polynomials by E J Barbeau informs me that "it is impossible to find four perfect squares in arithmetic progression"
Maybe you meant "the sum of terms in any initial segment is a perfect square" ?
11@ prophet sir, actually the 1st qsn is from http://www.artofproblemsolving.com/Forum/viewtopic.php?f=513&t=350686.
CMI-2010 qsn. [2]
@dmaga, a better soln, if possible.
341Look at Post 10: )Prove that there can't exist a infinite A.P. all of whose terms are perfect square. He has given the link to a discussion about the theorem I quoted.
http://www.mathpages.com/home/kmath044/kmath044.htm
In fact the book by Barbeau gives an example for 3 squares, 1,25,49. Then he goes on to state there are no APs with four squares as consecutive terms.
