Praise be to Allah.
The scholars (may Allah have mercy on them) differed concerning this issue. The correct view is that it is sufficient to do one prostration, so as to avoid hardship and difficulty. So one should prostrate when reciting the verse of prostration for the first time, then not repeat the prostration again.
This is the view of the Hanafis, and some of the Shaafa‘is and Hanbalis. It is also the view favoured by Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him).
Ibn ‘Aabideen (may Allah have mercy on him) said:
if he repeats it in two sitting, he should repeat the prostration, but if he repeats (the verse) in one sitting he does not have to repeat the prostration; rather once is sufficient, and doing it after the first recitation is preferable. This is so as to avoid hardship so long as the same verse is repeated in the same sitting.
End quote from Radd al-Muhtaar ‘ala ad-Durr al-Mukhtaar, 2/114. See also al-Insaaf, 2/196
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: Is it obligatory for the one who is reading Qur’aan to prostrate when he comes to a verse in which there is a sajdah (prostration)? If he is repeating the verse for the purpose of memorisation, does he have to prostrate every time?
He replied: He does not have to prostrate, whether he reads the verse in which there is a prostration once or he repeats it. The prostration of recitation is Sunnah and not obligatory. The evidence for that is the fact that ‘Umar ibn al-Khattaab (may Allah be pleased with him) recited in a Friday khutbah a verse in which there is a prostration, namely the verse in Soorat an-Nahl, and he came down (from the minbar) and prostrated, then he recited the same verse on another Friday, and he did not prostrate. Then he said: Allah has not made the prostration obligatory upon us unless we want to do it. So the prostration of recitation is Sunnah and is not obligatory.
If the verses are repeated, such as when one repeats the verses for the purpose of memorising Qur’aan, doing the prostration the first time will suffice for the rest and there is no need to repeat the prostration. But if he is reading Soorat al-Hajj, for example, (in which there are two sajdahs) and he does the first prostration then he comes to the second prostration, he may prostrate again even if the interval was not long.
End quote from Majmoo‘ al-Fataawa, 14/318
And Allah knows best.