Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem C. 1471. (March 2018)

C. 1471. Prove that every power of two greater than four can be expressed as the difference of two odd square numbers. For example, $\displaystyle 32=81-49$.

(5 pont)

Deadline expired on April 10, 2018.

Sorry, the solution is available only in Hungarian. Google translation

Megoldás. Az állítás: $\displaystyle 2^k=a^2-b^2=(a-b)(a+b)$, ahol $\displaystyle a$ és $\displaystyle b$ páratlan számok. Mivel $\displaystyle (a-b)$ és $\displaystyle (a+b)$ szorzata kettő-hatvány, ezért $\displaystyle (a-b)$ és $\displaystyle (a+b)$ is kettő hatványai.

Legyen $\displaystyle a-b=2$, vagyis $\displaystyle a=b+2$. Ekkor

$\displaystyle 2^k=2(2b+2),$

$\displaystyle 2^{k-1}=2b+2,$

$\displaystyle b=\frac{2^{k-1}-2}{2}=2^{k-2}-1,$

$\displaystyle a=b+2=2^{k-2}+1.$

Tehát bármely $\displaystyle k>2$ egész számra, vagyis $\displaystyle 2^k>4$ esetén $\displaystyle a=2^{k-2}+1$ és $\displaystyle b=2^{k-2}-1$ olyan páratlan számok, melyekre

$\displaystyle a^2-b^2=(2^{k-2}+1)^2-(2^{k-2}-1)^2=$

$\displaystyle 2^{2k-4}+2\cdot2^{k-2}+1-2^{2k-4}+2\cdot2^{k-2}-1=4\cdot2^{k-2}=2^k.$

### Statistics:

 138 students sent a solution. 5 points: 76 students. 4 points: 12 students. 3 points: 13 students. 2 points: 4 students. 1 point: 11 students. 0 point: 12 students. Unfair, not evaluated: 10 solutionss.

Problems in Mathematics of KöMaL, March 2018