educationelementary-number-theorypuzzle
I am going through some simple math problems in these quarantine days. Some of them are solved by me and some of them aren't solved by me. one of the unsolvable problems is given here:
Can you help me by giving any clue to fund the unknown number in the image?
The smallest infinitely often occurring prime gap, or
$$\liminf_{n\to\infty}\; (p_{n+1} - p_n)$$
is unknown as of now. Most likely, it is $2$, but the twin prime conjecture has not yet been settled.
Due to the work of Yitang Zhang and subsequent improvements by others (notably James Maynard and Terence Tao), we know that some prime gaps occur infinitely often. Zhang proved that gaps not larger than 70 million occur infinitely often, and the improvements lowered the bound to $246$ (perhaps there have been recent further improvements I'm not aware of).
I have drawn your triangle, but the sides are named with lower case letters instead of capitals thanks to Geogebra. Side $a$ is $10 \tan (70^\circ)$ , then you can get $c$ from either $c=\sqrt {a^2+b^2}$ or $c=\frac {10}{\cos (70^\circ)}$ If you have a right angle, finding the coordinates of $B$ is pretty easy. If you don't, you need to solve a pair of simultaneous equations that represent the lengths from A to B and C to B
Best Answer
Let the missing number be $x$ $$9=(10-7)^2$$ $$25=(15-10)^2$$ $$49=(15-8)^2$$ Therefore $$x={(8-7)^2}=1$$