Are you really sure?

Algebra Level 2

1 + 7 ( 0 × 1 × 2 × 3 ) = 1 2 1 + 7 ( 1 × 2 × 3 × 4 ) = 1 3 2 1 + 7 ( 2 × 3 × 4 × 5 ) = 2 9 2 \begin{aligned} 1 + 7(0 \times 1 \times 2 \times 3) & = 1^2 \\ 1 + 7(1 \times 2 \times 3 \times 4) & = 13^2 \\ 1 + 7(2 \times 3 \times 4 \times 5) & = 29^2 \end{aligned}

Is 1 plus 7 times the product of four consecutive integers always a perfect square?

Inspiration

No Yes

Arjen Vreugdenhil by Arjen Vreugdenhil , 44, USA

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2 solutions

Arjen Vreugdenhil
May 18, 2018

The next term in the sequence is 1 + 7 ( 3 × 4 × 5 × 6 ) = 2521 , 1 + 7(3\times 4 \times 5 \times 6) = 2521, which is not a perfect square.

The answer is therefore No \boxed{\text{No}} .

In fact, the next term that does product a perfect square is 1 + 7 ( 34 × 35 × 36 × 37 ) = 333 1 2 . 1 + 7(34\times 35\times 36\times 37) = 3331^2.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Also 1 + 7 ( 10 ∗ 11 ∗ 12 ∗ 13 ) = 120121 1 + 7 (10*11*12*13) = 120121 is not aperfect square !!

Ossama Ismail - 3 years ago

What is the significance of this?

Zachary Berry - 2 years, 2 months ago
Ritabrata Roy
Jul 23, 2018

1+7(5×6×7×8)≠square

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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