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August 6, 2019

Belarus MO 2018 Problem 10.5 - Number Theory

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$has a solution in integers $a$ and $b$.

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Start with hints

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Belarus MO 2018 Problem 10.5 [/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.26.6" open="on"]Number Theory [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.26.6" open="off"]5/10 [/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.26.6" open="off"]An Introduction to Number Theory [/et_pb_accordion_item][/et_pb_accordion][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.6" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.6"]Let's check for n = 1. Observe that a = 27, b = 13 gives a solutions for n = 1. What about higher degrees? Can we use this information?  [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.6"]Does it work for n = 2? Let's prove something general! Prove that for a, b to have solutions, n must be odd.  [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.6"]If n is even, Take $\pmod{3}$ to see that $-b^2\equiv 1\pmod{3}$, which has no integer solutions in $b$ Hence, n must be odd. [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.6"]Well now take n odd. Say $n=2m+1$ for some positive integer $m$. Then, the solution $(a,b)=(27\times 2018^m, 13\times 2018^m)$ exists and works. [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.26.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Watch video

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Connected Program at Cheenta

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Similar Problems

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2 comments on “Belarus MO 2018 Problem 10.5 - Number Theory”

    1. Which answers? They are provided in the last hint always. All the best. Hope you are loving the Sequential Hints Method!

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