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June 8, 2019

A sequence of (mostly) prime numbers

[et_pb_section fb_built="1" _builder_version="3.22.4"][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_text _builder_version="3.22.4" 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_padding="20px|20px|20px|20px"]

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 pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$, $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms.

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.22.4"][et_pb_column type="4_4" _builder_version="3.22.4"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.23.3" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Bulgarian Mathematical Olympiad 2002[/et_pb_accordion_item][et_pb_accordion_item title="Topic" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Number theory[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" open="off" _builder_version="3.23.3" title_text_shadow_horizontal_length="0em" title_text_shadow_vertical_length="0em" title_text_shadow_blur_strength="0em" closed_title_text_shadow_horizontal_length="0em" closed_title_text_shadow_vertical_length="0em" closed_title_text_shadow_blur_strength="0em"]Problem Solving Strategies by Arthur Engel[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" 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"]

Start with hints

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[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]There exists a general theory of second-order linear difference equations. Read about it here.[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]Can we have a_n>a_{n+1} for all n? What happens if we do? What happens otherwise?

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"]Note that if a_n<a_{n+1} for some n then the sequence becomes increasing thereafter. Use this fact to simplify the recurrence relation and solve it explicitly.

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The sequence cannot be decreasing because it is a sequence of positive integers. Hence there exists (a smallest) k such that a_k\le a_{k+1}. If a_k=a_{k+1} then the sequence becomes constant from the kth term onwards (we shall treat this case later). Otherwise 3a_{k+1}-2a_k>a_k hence a_{k+2}>a_{k+1}. This implies that the sequence becomes increasing from the kth term onwards. Also, a_{n+2}=3a_{n+1}-2a_n for n\ge k. This difference equation has the characteristic equation \lambda^2-3\lambda +2=0 (see the link in hint 1) which has the solutions \lambda = 2,1. Thus, a_{n+k}=2^nA+B for A,B satisfying A+B=a_k, 2A+B=a_{k+1}. Take any prime divisor p of A+B. By Fermat's little theorem, 2^{m(p-1)} \equiv 1 \; (\text{mod}\; p) for every positive integer m. Thus 2^{m(p-1)}A+B\equiv A+B\equiv 0\; (\text{mod}\; p). Hence the sequence contains infinitely many composites. This cannot be allowed, so the sequence cannot be strictly increasing at any point.  

The above discussion shows that, for any permissible sequence, there exists a (smallest) j and a prime q such that a_n=q for all n\ge j. For n<j, the sequence is decreasing. Note that, either q=a_{j+1}=3a_j-2a_{j-1}=3q-2a_{j-1} or q =2a_{j-1}-3q. Hence, either a_{j-1}=q or a_{j-1}=2q. The first one can happen only if j=1 because otherwise the minimality of j is violated. In that case, the sequence is constant and b=c=q. If a_{j-1}=2q then either j=2 (in which case b=2q, c=q) or q=|6q-2a_{j-2}| hence a_{j-2}=3q\pm\frac{q}{2}. The last equality forces q to be 2. Thus a_{j-2}=6\pm 1. If j>3 then 4=a_{j-1}=|18\pm 3 - 2a_{j-3}| which is absurd as 2a_{j-3} cannot be an odd number. Hence j=3 in this case and c=4,b=5,7.

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Watch the video (Coming Soon)

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Similar Problems

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