Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
Brother, thanks for this serious question, which reflects your concern about Islam. We commend your efforts in pursuit of truth. This is what is required of all people, to seek the truth and not to give in to anything that may cloud their minds with false ideas about Islam.
In his response to your question, Dr. Muzammil H. Siddiqi, former President of the Islamic Society of North America, states:
“In the Qur'an, Allah, Most High, has given us some specific times of prayers, but He did not mention the number. It is not said anywhere in the Qur'an that one must pray five times or three times. The Qur'an mentions that the prayer is prescribed for the believers at specific times (An-Nisa': 103).
In Surat Al-Isra' Allah says, "Establish regular prayers at the sun's decline till the darkness of the night, and (the recital of) the Qur’an at dawn. Lo! (the recital of) the Qur’an at dawn is ever witnessed. And some part of the night awake for it, a largess for thee. It may be that thy Lord will raise thee to a praised estate.” (Al-Isra’: 78-79)
The time of “the sun's decline” (duluk ash-shams) is interpreted as Zhuhr, `Asr and Maghrib. The "darkness of the night" (ghasaq al-layl) is interpreted as `Isha’ prayer. And "the recital of the Qur’an at dawn " (Qur'an al-fajr) is Fajr prayer.
In Surat Taha Allah says, "Therefore be patient with what they say, and celebrate (constantly) the praises of thy Lord, before the rising of the sun, and before its setting; yea, celebrate them for part of the hours of the night, and at the sides of the day: that thou mayest have (spiritual) joy." (Taha: 130)
In Surat Ar-Rum Allah says, "So (give) glory to Allah, when you reach eventide and when you rise in the morning. Yea, to Him be praise, in the heavens and on earth; and in the late afternoon and when the day begins to decline.” (Ar-Rum: 17-18)
The number of prayers are not mentioned in the Qur'an, but they are mentioned in the hadiths of the Prophet (peace and blessings be upon him). The Prophet (peace and blessings be upon him) under the guidance of Allah told us when to pray, how many times to pray and how to pray. He gave us five times of prayers. He told us that Allah has prescribed for us five prayers every day. There are hundreds of hadiths on this subject. The whole Ummah is unanimous and has a consensus (ijma`) that five prayers are obligatory. Anyone who denies this or says that the obligatory prayers are less than five or more than five, that person is not a Muslim.
Your cousin either does not know what Islam is, or someone has misled him or he is trying to mislead you. Have no doubt in your mind that five prayers are obligatory and they must be performed.”
If you are still in need of more information, don't hesitate to contact us. Do keep in touch. May Allah guide us all to the straight path!