Tuesday, March 27, 2018.
This is about my alternate take on: Solve an equation with integer variables
We have integer sequences $x_n, y_n$ with $x_0 = 0, x_1 = 16, x_2 = 552, x_3 = 18760,$ continuing $$ x_{n+2} = 34 x_{n+1} - x_n + 8 \; \; . $$ We also have $y_0 = 1, y_1 = 23, y_2 = 781, y_3 = 26531,$ continuing $$ y_{n+2} = 34 y_{n+1} - y_n \; \; , $$ with no constant term for $y.$ The result is that we have the sequence of even $x$ such that $$ 2 x^2 + x + 1 = y^2 $$
The original question asked to show that $x$ can be a power of two only for $x=16.$ My take was that the same conclusion can be realized if the 2-adic valuation of $x_n$ grows slowly. Indeed, in the two output below, when $n = 2^w,$ we see that $x_n = 2^{w+2} \cdot \mbox{ODD} \; \; .$ In the first section of output, we see that $n=2^w$ does fairly well, it does not always set a new record for the power of 2 dividing $x_n,$ but nearly so.
QUESTION: Can we prove a good upper bound for the power of two $e$ dividing $x_n,$ meaning $2^e \parallel x_n \; ,$ inequality suspected to be $$ e < n + 5 \; \; \; ? $$ Note that many weaker inequalities will solve the original question, such as $$ e < 3 n + 5 \; \; \; ? $$ because this will still rule out $x_n$ being a power of 2 unless $n$ is small.
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x_{n+2} = 34 x_{n+1} - x_n + 8. y_{n+2} = 34 y_{n+1} - y_n.
2 x^2 + x + 1 = y^2 . y_0 = 1; y_1 = 23; y_2 = 781;
2 * 0^2 + 0 + 1 = 1 = 1^2
2 * 16^2 + 16 + 1 = 529 = 23^2
2 * 552^2 + 552 + 1 = 609961 = 781^2
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Tue Mar 27 10:46:16 PDT 2018
2 552 = 2^3 3 23
3 18760 = 2^3 5 7 67
4 637296 = 2^4 3 11 17 71
5 21649312 = 2^5 29 41 569
8 848696338272 = 2^5 3 17 cdot mbox{BIG}
13 38394236450627331136 = 2^6 23 79 cdot mbox{BIG}
16 = 2^6 3 11 17 cdot mbox{BIG}
29 = 2^7 cdot mbox{BIG}
32 = 2^7 3 17 cdot mbox{BIG}
61 = 2^9 cdot mbox{BIG}
128 = 2^9 3 17 cdot mbox{BIG}
189 = 2^10 5 7^2 13^2 23 53 cdot mbox{BIG}
256 = 2^10 3 11 17 cdot mbox{BIG}
445 = 2^11 29 41 67 cdot mbox{BIG}
512 = 2^11 3 17 43 cdot mbox{BIG}
957 = 2^13 5 7 23 cdot mbox{BIG}
2048 = 2^13 3 17 23 cdot mbox{BIG}
3005 = 2^17 23 29 41 cdot mbox{BIG}
32768 = 2^17 3 17 37 cdot mbox{BIG}
35773 = 2^18 cdot mbox{BIG}
65536 = 2^18 3 11 17 cdot mbox{BIG}
101309 = 2^19 79 cdot mbox{BIG}
131072 = 2^19 3 17 cdot mbox{BIG}
232381 = 2^22 cdot mbox{BIG}
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Tue Mar 27 10:58:21 PDT 2018
2 552 = 2^3 3 23
4 637296 = 2^4 3 11 17 71
8 848696338272 = 2^5 3 17 cdot mbox{BIG}
16 = 2^6 3 11 17 cdot mbox{BIG}
32 = 2^7 3 17 cdot mbox{BIG}
64 = 2^8 3 11 17 cdot mbox{BIG}
128 = 2^9 3 17 cdot mbox{BIG}
256 = 2^10 3 11 17 cdot mbox{BIG}
512 = 2^11 3 17 43 cdot mbox{BIG}
1024 = 2^12 3 11 17 cdot mbox{BIG}
2048 = 2^13 3 17 23 cdot mbox{BIG}
4096 = 2^14 3 11 17 cdot mbox{BIG}
8192 = 2^15 3 17 53 cdot mbox{BIG}
16384 = 2^16 3 11 17 71 cdot mbox{BIG}
32768 = 2^17 3 17 37 cdot mbox{BIG}
65536 = 2^18 3 11 17 cdot mbox{BIG}
131072 = 2^19 3 17 cdot mbox{BIG}
262144 = 2^20 3 11 17 cdot mbox{BIG}
524288 = 2^21 3 17 43 cdot mbox{BIG}
1048576 = 2^22 3 11 17 cdot mbox{BIG}
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Here is some original output, edited for clarity. $x_n = (w-1)/4, \;$ and is an integer only when $w \equiv 1,5 \pmod 8,$ then is even only when $w \equiv 1 \pmod 8 \; .$
jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
Automorphism matrix:
3 8
1 3
Automorphism backwards:
3 -8
-1 3
3^2 - 8 1^2 = 1
w^2 - 8 v^2 = -7
Tue Mar 27 11:51:16 PDT 2018
w: 1 v: 1 SEED KEEP +- w mod eight: 1 x_n = 0 =
w: 5 v: 2 SEED BACK ONE STEP -1 , 1 w mod eight: 5 x_n = 1 = 1
w: 11 v: 4 w mod eight: 3
w: 31 v: 11 w mod eight: 7
w: 65 v: 23 w mod eight: 1 x_n = 16 = 2^4
w: 181 v: 64 w mod eight: 5 x_n = 45 = 3^2 5
w: 379 v: 134 w mod eight: 3
w: 1055 v: 373 w mod eight: 7
w: 2209 v: 781 w mod eight: 1 x_n = 552 = 2^3 3 23
w: 6149 v: 2174 w mod eight: 5 x_n = 1537 = 29 53
w: 12875 v: 4552 w mod eight: 3
w: 35839 v: 12671 w mod eight: 7
w: 75041 v: 26531 w mod eight: 1 x_n = 18760 = 2^3 5 7 67
w: 208885 v: 73852 w mod eight: 5 x_n = 52221 = 3 13^2 103
w: 437371 v: 154634 w mod eight: 3
w: 1217471 v: 430441 w mod eight: 7
w: 2549185 v: 901273 w mod eight: 1 x_n = 637296 = 2^4 3 11 17 71
w: 7095941 v: 2508794 w mod eight: 5 x_n = 1773985 = 5 197 1801
w: 14857739 v: 5253004 w mod eight: 3
w: 41358175 v: 14622323 w mod eight: 7
w: 86597249 v: 30616751 w mod eight: 1 x_n = 21649312 = 2^5 29 41 569
w: 241053109 v: 85225144 w mod eight: 5 x_n = 60263277 = 3 3499 5741
w: 504725755 v: 178447502 w mod eight: 3
Tue Mar 27 12:00:17 PDT 2018
w^2 - 8 v^2 = -7
jagy@phobeusjunior:~$
jagy@phobeusjunior:~$