Acer Aspire 51 won't boot until I took out the battery

I have an Acer Aspire 5100 laptop. It would boot up and shut down after a few seconds. I put it on a hard surface and it started working for awhile. Then after about a few week it started to boot and shut down after a few seconds again.
I cleaned the memory, took out the battery and the power cord and pressed the power button for 58 seconds and put things back together but it still didn't work.
I took the battery out and plugged in the power cord while the battery was out and pushed the power button and it worked. Then I put the battery back in and it boot up a few seconds and shut down. I took the battery out again and it boots up normally.
My question is: Can I continue to run it without the battery? Will it harm the computer? Do I need to get another battery? The battery is a Li-ion Rechargeable Model BATBL50L6, Charging Current 3.2A.
If this is a solution to a problem and it is safe, I want everyone to know.