Find minimum 2 digit n, such that \((7^{n} + 7^{n-1} + 7^{n-2} + ...... + 7^{3} + 7^{2} + 7^{1} + 7^{0})\) is a perfect square, since known \((7^{3} + 7^{2} + 7^{1} + 7^{0})\) is a perfect square.

**Note :** \(n \neq 3\) !!!

Find minimum 2 digit n, such that \((7^{n} + 7^{n-1} + 7^{n-2} + ...... + 7^{3} + 7^{2} + 7^{1} + 7^{0})\) is a perfect square, since known \((7^{3} + 7^{2} + 7^{1} + 7^{0})\) is a perfect square.

**Note :** \(n \neq 3\) !!!

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TopNewestI am sorry, bcs. I have no answer. – Bryan Lee Shi Yang · 2 years, 2 months ago

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here .!!! @Bryan Lee Shi Yang – Sandeep Bhardwaj · 2 years, 2 months ago

If you take \(n=0\), then you expression will be equal to \(1\) which is a perfect square. So \(n=0\) can be the one value. Also try to use the latex coding in your notes and problems (now I've put latex in it) , if you need a guide for latex, you can find it just by clickingLog in to reply

CAN U WRITE A PROOF TO SUPPORT THIS – Vishwesh Agrawal · 2 years, 2 months ago

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There is no solution < 270. – D G · 2 years, 2 months ago

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– Bryan Lee Shi Yang · 2 years, 2 months ago

Are there any solutions bigger than 270?Log in to reply

Up till n=22, only n=0 and n=4 are perfect square according to what I get from TI -83+ calculator. After that I do not get accurate calculations. – Niranjan Khanderia · 2 years, 2 months ago

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\(n = 37\) satisfies! – Kartik Sharma · 2 years, 2 months ago

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My code! *Easy code, though! – Kartik Sharma · 2 years, 2 months ago

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– D G · 2 years, 2 months ago

No it doesn't. 21655801907853836686195357616408 is not a square number.Log in to reply

– Kartik Sharma · 2 years, 2 months ago

\(\frac{{7}^{38}-1}{6} = 21655801907853831608163177358336 = {4653579472605344}^{2}\)Log in to reply

– Bryan Lee Shi Yang · 2 years, 2 months ago

Sorry, it is +1, not -1.Log in to reply

– Kartik Sharma · 2 years, 2 months ago

Oh maybe. My IDLE is just reversing it up. When I replace - with +, it shows the above value(which should come with -).Log in to reply

– Siddhartha Srivastava · 2 years, 2 months ago

http://www.wolframalpha.com/input/?i=%287%5E38+-+1%29%2F6+&dataset=Log in to reply

– Kartik Sharma · 2 years, 2 months ago

There has to be a problem with my python idle, then. Thanks!Log in to reply