Blood pressure readings are typically different in left and right arms most because the blood vessel anatomy is not quite as nice and symmetrical as things on the outside (eyes, ears, arms, etc.). So, asymmetry results in somewhat different blood paths and resultant blood pressures. For the most part, a difference in systolic blood pressure (the top number) that is 'acceptable' is about 10-15mmHg. Above that, particularly in someone with a diagnosis of high blood pressure, there might be concern that one of the blood vessels is (the side with the lower blood pressure) has an unusually tight narrowing (beyond normal anatomy). This could be due to several conditions, some you are born with and others that can develop over time like plaque build up. The only way to know is to look at the blood vessels with some sort of imaging study (ultrasound, CT scan, MRI) if it is of concern. Clinically, the presence of one narrowed blood vessel could mean other blood vessels are narrowed elsewhere in the body. As far as treatment of blood pressure in these settings, we usually treat the higher reading and stick with that limb. I would not be fooled into thinking that because the blood pressure is "normal " on one side of the body that you should stop your medication. Remember that your brain and heart are likely feeling the effects of the higher blood pressure--not the lower.