Need help with LM317 (500 mA) out of 3 volts supply Calculations
No.

LM317 has a dropout voltage of about 3 V. That means the output pin of LM317 must be at least 3 V below the input pin for proper regulation.

Furthermore, in this configuration the output node of the current source is 1.25 V below the output pin of the LM317 chip.

So the input must be at least 4.25 V to provide a regulated 500 mA output. And then it will only do so into a short circuit. The input voltage must be further increased by whatever you want the output compliance voltage to be if you're driving a load that produces any voltage drop with current running through it.

To put it in an equation, you need to have 

$$V_{in} \ge V_{Output} + 4.25\ {\rm V}$$

for this circuit to maintain regulation.