How to authenticate the mobile number 

In the normal authentication flow, we use SMS to send OTP and Push Notification to get same OTP code to validate. But I think you want to send SMS without any SMS gateway. So you can try to write a java program that will send a SMS by using modem that have facility to send SMS like internet modem.

I found an API that provide Online validation of Phone number, Try this one: http://www.phone-validator.net/phone-number-online-validation-api.html

But you need your work Email id, not possible with free email address.

Reference:

Googling