The data sheet states that Adjustment Pin Current is typically 50 and max 100 micro Amps. 1.25 / 100 micro amps = 12.5K, so 500 is well within the operating area. 120 and 240 ohms are convenient standards, but not the rule, IMO. I don't suppose you would believe I have a very rare zener LED would you? No? OK you're right, It's drawn in backwards, Thanks.
EDIT I see you strongly believe in 120 ohms for R1, Think you'll ever change your mind and accept that it can be something else? http://www.electro-tech-online.com/threads/lm317-calculating.119697/
Yes, I did a little work with 400Hz inverters, the theory behind 400 Hz is quite interesting, if I remember right, the ideal frequency for motors is 380 Hz, so they rounded it up to 400 Hz, a motor running at 400 Hz can produce 3 time the HP of the same size and weight motor running at 60 Hz.
The negative side won't get used much and just reducing the ripple will be ok for what I do. I will probably just set the voltage around 16 and forget about it.
Just so you know, the minimum resistance on the output is NOT based on the adjustment pin current. It is based on the minimum OUTPUT current that should be drawn from the device so that it maintains regulation. Below a certain OUTPUT current (not adjustment pin current) the output could go up somewhat and that means the output voltage is no longer regulated. That's why they always show a somewhat low value like 220 ohm at the output of the LM317 regulators. But 100 ohms is better because it covers more devices under more operating conditions, so that's why such a low value is recommended.
There is one time when this is not necessary, but you have to be careful here. This is when the output will ALWAYS be loaded by the following circuit at more than something like 10ma. If your circuit always draws 10ma then you can use other resistor values that are somewhat higher like you wanted to do, but you have to be very careful here because the following circuit may NORMALLY draw 10ma but during startup it may only draw 1ma for example, and if that happens then during startup the following circuit will receive an over voltage condition which could burn it out and then it wont work at all.
The 100 ohmish resistor is chosen to always draw around 10ma or so and that keeps the min output current spec happy regardless of the following circuit current draw or how that current changes over the operating time of the product.
About LED's and zeners...
LED's do act like crummy zeners because their operating curve looks a little like zeners. I used an LED as a zener in a circuit long time ago when i realized that powering an LED from an unknown voltage source that can vary quite a bit would change the brightness of the LED quite a bit when using just a series resistor alone.
The LED acts a lot like a zener as it does 'regulate' the voltage to some degree when the current is limited to something the LED can easily handle. But when done it is not connected backwards, it is connected as a normal LED.