Live Long, the first part, means to have a life that is not cut short, that it will be long enough for us to reach all the goals and achieve all the things we have set out for; that we won't just be blasted by a ticked off Romulan. Prosper, a very meaningful word. Many people have differences of opinions on what it means to prosper. A lot of people apply it to money. To prosper, you must be rich, and be able to afford anything. Others may think of it as success. They may have prospered in their field. Prospering means to be successful or fortunate; thrive; flourish. However one views being fortunate or successful, that is the Prosper implied by the Vulcan salute. So, when Spock raises his hand, pointer and middle finger together, ring finger and pinky put together, split between the middle and ring, thumb extended, he is wishing you to have a long, full life in which you have time to achieve your desires, and to be successful and flourish in all of your endeavors... But of course, that's a bit of a mouthful, so it can be summed up simply in "Live Long and Prosper."
Vulcans are an interesting race. They thrive on being stoic and emotionless. Their intelligence is almost unmatched, and their ability to stay calm in even the most stress filled situations make them a valuable asset. Spock differs from most of his planet, as he comes from a union between Vulcan and human, and is often looked down upon for it. He has several characteristics of humanity, including a temper and compassion, unusual of Vulcans.
Now, if someone such as this wants those he meets to have such a wonderful life, how much more would our Heavenly Father want the same? Moses 1:39 states "For behold, this is my work and my glory- to bring to pass the immortality and eternal life of man." Sounds an awful lot like "Live Long and Prosper", doesn't it? That's exactly what God wants for all of us. He wants us to have a long, productive life in which we can live a good life, and follow His commandments, to return to live with Him. There's no way to be more prosperous than to have Eternal Life, to live exalted in the presence of our Heavenly Father, and His son, Jesus Christ. Our Heavenly Father wants us to be happy; He wants us to succeed and to flourish. It is His greatest desire that we achieve this, and His very purpose to make it happen! He cares so much for us, and would like nothing more than to have all of His children, happy and smiling, satisfied with their mortal lives, to return to live with Him forever in eternal glory. So, next time you read Moses 1:39, think of it as God giving you the Vulcan salute. Live Long and Prosper.