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 Quarting the cube (Posted on 2013-02-28)

Select integer x and triangular number y such that 8y=3x^4-2x^2-1.

Prove that y is divisible by 28 - or find a counter-example.

 See The Solution Submitted by broll Rating: 5.0000 (2 votes)

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 SOLUTION +some add-ons | Comment 3 of 6 |

Counterexample? My dear broll, you must be kidding...

Not only that the  "y" s  of the (x,y) couples defined by your   formula

are divisible by 28,-  so are all the "y"s generated by an odd integer x. As long as    x mod 7 = 4,6,1, or 3 – no one forces y to be  a triang. number.

As a matter of fact 4/7 of the "y"s generated by an odd integer are divisible by 28, 2/7 are divisible by 4 and 1/7 is divisible by 2. -see the table below , showing  few "y"s corresponding to odd integers:
(the mod 28 values cycle is 4,20,4, 0,0,0,0,   etc )

<table class="MsoNormalTable" dir="rtl" border="0" cellspacing="0" cellpadding="0" style="border-collapse:collapse;mso-table-layout-alt:fixed;mso-padding-alt: 0cm 1.5pt 0cm 1.5pt;mso-table-dir:bidi"> <tbody><tr style="mso-yfti-irow:0;mso-yfti-firstrow:yes;height:17.3pt"> <td width="73" valign="top" style="width:55.1pt;padding:0cm 1.5pt 0cm 1.5pt; height:17.3pt">

ymod28<o:p></o:p>

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y<o:p></o:p>

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x<o:p></o:p>

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0<o:p></o:p>

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0<o:p></o:p>

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1<o:p></o:p>

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0<o:p></o:p>

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28<o:p></o:p>

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3<o:p></o:p>

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4<o:p></o:p>

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228<o:p></o:p>

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5<o:p></o:p>

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20<o:p></o:p>

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888<o:p></o:p>

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7<o:p></o:p>

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4<o:p></o:p>

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2440<o:p></o:p>

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9<o:p></o:p>

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0<o:p></o:p>

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5460<o:p></o:p>

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11<o:p></o:p>

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0<o:p></o:p>

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10668<o:p></o:p>

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13<o:p></o:p>

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0<o:p></o:p>

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18928<o:p></o:p>

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15<o:p></o:p>

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0<o:p></o:p>

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31248<o:p></o:p>

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17<o:p></o:p>

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4<o:p></o:p>

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48780<o:p></o:p>

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19<o:p></o:p>

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20<o:p></o:p>

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72820<o:p></o:p>

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21<o:p></o:p>

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4<o:p></o:p>

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104808<o:p></o:p>

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23<o:p></o:p>

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0<o:p></o:p>

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146328<o:p></o:p>

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25<o:p></o:p>

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0<o:p></o:p>

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199108<o:p></o:p>

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27<o:p></o:p>

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0<o:p></o:p>

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265020<o:p></o:p>

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29<o:p></o:p>

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0<o:p></o:p>

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346080<o:p></o:p>

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31<o:p></o:p>

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4<o:p></o:p>

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444448<o:p></o:p>

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33<o:p></o:p>

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20<o:p></o:p>

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562428<o:p></o:p>

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35<o:p></o:p>

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4<o:p></o:p>

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702468<o:p></o:p>

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37<o:p></o:p>

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0<o:p></o:p>

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867160<o:p></o:p>

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39<o:p></o:p>

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0<o:p></o:p>

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1059240<o:p></o:p>

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41<o:p></o:p>

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0<o:p></o:p>

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1281588<o:p></o:p>

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43<o:p></o:p>

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0<o:p></o:p>

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1537228<o:p></o:p>

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45<o:p></o:p>

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4<o:p></o:p>

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1829328<o:p></o:p>

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47<o:p></o:p>

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</td> </tr> </tbody></table>

As you can notice  triangle numbers are a subset of the y column:    (0, 28, 5460, 1059240)   etc  correspond to  x column values (1, 3, 11, 41     etc)

For x sequence to continue  I have assumed a linear recurrence x(n)= 4*x(n-1)- x(n-2)
(similar to  "fun sequence"  !!) …       and got

x(5)=153 ;   x(6) = 459 ,  compared with Charlie's table, got the corresponding "y" s and assumed a linear recurrence
y(n)= A*y(n-1)+B*y(n-2)+C , then  aided by a corresponding applet got the equations constants :
y(n)= 194*y(n-1)1-y(n-2)+28.

Since both y(1) and y(2) i.e. 0 and 28 are divisible by 28, and     both 194 and 28

are divisible by 28  -      all the triangle numbers in the y sequence will be 0 mod 28.

BTW:   all the triangle numbers related to this problem are 4/ 3 of a valid triangle number-

(0 vs 0), (28 vs 21),(5460 vs 4095),

and the x values are such that 3*x^2-2 is a square.

Both the x sequence and y sequence are cited in the OEIS  under  A001835  &  A200999 correspondingly.

<o:p> </o:p>

<o:p> </o:p>

 Posted by Ady TZIDON on 2013-03-01 21:18:12

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