MathLabs

Problem 4

Given a set MM of 19851985 distinct positive integers, none of which has a prime divisor greater than 2323, prove that MM contains four distinct elements whose product is the fourth power of an integer.
Step 4 of 4: The four selected numbers give a fourth power
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In plain words

Two square products whose square roots also pair to a square combine into one fourth power; the four original numbers are guaranteed distinct because the pairs were disjoint.

xpypxqyq=sp2sq2=(spsq)2=(u2)2=u4x_py_p x_qy_q=s_p^2s_q^2=(s_ps_q)^2=(u^2)^2=u^4
Detailed analysis

The two pairs are disjoint, so xp,yp,xq,yqx_p,y_p,x_q,y_q are four distinct elements of MM. Their product is xpypxqyq=sp2sq2=(spsq)2=u4x_py_p x_qy_q=s_p^2s_q^2=(s_ps_q)^2=u^4, a fourth power. This proves the claim.