Discrete math number problem. How would I justify the following statement. Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.

glyperezrl 2022-07-18 Answered
Discrete math number problem
How would I justify the following statement.
Two integers are consecutive if and only if one is more than the other. Any product of four consecutive integers is one less than a perfect square.
I think this is true.
because for example
2 < 3 < 4 < 5
2 4 5 3 = 120
Which one less than 121 a perfect square.
So how would I justify it I did
Let n be a integer
n ( n + 1 ) ( n + 2 ) ( n + 3 ) + 1 = ( m ) 2
But I am not sure how to proceed.
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Answers (2)

juicilysv
Answered 2022-07-19 Author has 17 answers
Explanation:
n ( n + 1 ) ( n + 2 ) ( n + 3 ) = n 4 + 6 n 3 + 11 n 2 + 6 n + 1 1 = n 2 ( n 2 + 6 n + 11 + 6 n + 1 n 2 ) 1 = n 2 ( ( n + 1 n ) 2 + 6 ( n + 1 n ) + 11 2 ) 1 = n 2 ( ( n + 1 n ) 2 + 6 ( n + 1 n ) + 9 ) 1 = n 2 ( n + 1 n + 3 ) 2 1 = ( n 2 + 3 n + 1 ) 2 1
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Macioccujx
Answered 2022-07-20 Author has 3 answers
Explanation:
Try to proceed heuristically: write down the numbers (possible squares) you get for the first few values of n. Write down the numbers they are the squares of. Try to find a relationship between this and n. Is the former a polynomial in n? Develop the product n ( n + 1 ) ( n + 2 ) ( n + 3 ).
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