MHB Proving $a^2-3a-19$ is Not Divisible by 289

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The discussion focuses on proving that the expression $a^2-3a-19$ cannot be divisible by 289. A proof by contradiction is employed, starting with the assumption that the expression is divisible by 289. It is shown that if $a^2-3a-19$ is divisible by $289$, then $(a+7)^2$ must also be divisible by 17, leading to the conclusion that $a+7$ must be a multiple of 17. Substituting this back into the expression reveals that it results in a form that is not divisible by 289, thus confirming that no integer $a$ satisfies the condition. The proof effectively demonstrates the impossibility of divisibility by 289 for the given expression.
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Show that there is no integer a for which $a^2-3a -19$ is divisible by 289
 
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kaliprasad said:
Show that there is no integer a for which $a^2-3a -19$ is divisible by 289
[sp]Proof by contradiction: Suppose that $a^2-3a -19$ is divisible by $289 = 17^2$. Since $a^2-3a -19 = (a+7)^2 -17(a+4)$, it follows that $(a+7)^2$ must be divisible by $17.$ But $17$ is prime, and therefore $a+7$ must be divisible by $17.$ So $a+7 = 17k$ for some integer $k$. But then $$a^2-3a -19 = (17k-7)^2 - 3(17k-7) - 19 = 289(k^2-k) + 51,$$ which is clearly not a multiple of $289$.[/sp]
 
kaliprasad said:
Show that there is no integer a for which $a^2-3a -19$ is divisible by 289

$$a^2-3a-19=289k$$

$$\Delta=85+1156k=17(5+68k)$$

For $\Delta$ to be a perfect square $$5+68k$$ must have an odd power of $17$ in its prime factorization and so $5+68k$ must be divisible by $17$, but we see

$$5+68k\equiv5\pmod{17}$$

and so $$a^2-3a-19$$ is not divisible by $289$.
 
2 good ans
here is mine

As a first step as we see that $289 = 17^2$

Now $a^2-3a-19 = (a-10)(a+7) + 51$

The 2nd term that is 51 is divisible by 17 and for the 1st term that is product
to be divisible by 17 either (a-10) or (a+7) is divisible by 17. but if one of them is
divisible by 17 then the 2nd one is also divisible by 17.

So 1st term is (a-10)(a+7) is divisible by 289 and 2nd term 51 is not divisible by 289
so sum is not divisible by 289. Or the 2nd term is divisible by 17 and 1st term is not
divisible by 17 so sum is not divisible by 17.
 
My solution (which is much like Opalg's):

Reducing mod $17$, and using the quadratic formula (since $2 \not\mid 17$):

$a^2 - 3a - 2 = 0$ (mod $17$), therefore:

$a = 2^{-1}[-(-3) \pm \sqrt{(-3)^2 - 4(-2)}]= 9(3) = 10$ (mod $17$).

Using $a = 17k + 10$ in the original equation, we obtain:

$(17k + 10)^2 - 3(17k + 10) - 19 = 289m$

$289k^2 + 289k + 51 = 289m \implies 17(m - k - k^2) = 3$

but $17$ does not divide $3$.
 
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