A number theory problem by Med Fatnassi
Find the number of quadruplets , where are integers, such that
The answer is 1.
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The proof is that we can find an upper bound for w, when w=x-1=y-1=z-1.
In this case for convenience we write it as w!=3t!, where w=t+1.
By definition of factorials we can express w! as wt!, so w=3 is its upper bound.
Indeed one solution set (3,2,2,2) exists.
w≤2 doesn't exist because 2! is 2, and all factorial are naturals, three of them must≥3.
One. QED