Le Verrier's most famous achievement is his prediction of the existence of the then unknown planet Neptune, using only mathematics and astronomical observations of the known planet Uranus.
Encouraged by physicist Arago, Director of the Paris Observatory, Le Verrier was intensely engaged for months in complex calculations to explain small but systematic discrepancies between Uranus's observed orbit and the one predicted from the laws of gravity of Newton.
At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England.
Le Verrier announced his final predicted position for Uranus's unseen perturbing planet publicly to the French Academy on 31 August 1846, two days before Adams's final solution was privately mailed to the Royal Greenwich Observatory.
Le Verrier transmitted his own prediction by 18 September in a letter to Johann Galle of the Berlin Observatory.
The letter arrived five days later, and the planet was found with the Berlin Fraunhofer refractor that same evening, 23 September 1846, by Galle and Heinrich d'Arrest within 1° of the predicted location near the boundary between Capricorn and Aquarius.

There was, and to an extent still is, controversy over the apportionment of credit for the discovery.
There is no ambiguity to the discovery claims of Le Verrier, Galle, and d'Arrest.
Adams's work was begun earlier than Le Verrier's but was finished later and was unrelated to the actual discovery.
Not even the briefest account of Adams's predicted orbital elements was published until more than a month after Berlin's visual confirmation.
Adams made full public acknowledgement of Le Verrier's priority and credit (not forgetting to mention the role of Galle) when he gave his paper to the Royal Astronomical Society in November 1846:
I mention these dates merely to show that my results were arrived at independently, and previously to the publication of those of M. Le Verrier, and not with the intention of interfering with his just claims to the honours of the discovery; for there is no doubt that his researches were first published to the world, and led to the actual discovery of the planet by Dr. Galle, so that the facts stated above cannot detract, in the slightest degree, from the credit due to M. Le Verrier.

— Adams (1846)
Le Verrier's most famous achievement is his prediction of the existence of the then unknown planet Neptune, using only mathematics and astronomical observations of the known planet Uranus. Encouraged by physicist Arago, Director of the Paris Observatory, Le Verrier was intensely engaged for months in complex calculations to explain small but systematic discrepancies between Uranus's observed orbit and the one predicted from the laws of gravity of Newton. At the same time, but unknown to Le Verrier, similar calculations were made by John Couch Adams in England. Le Verrier announced his final predicted position for Uranus's unseen perturbing planet publicly to the French Academy on 31 August 1846, two days before Adams's final solution was privately mailed to the Royal Greenwich Observatory. Le Verrier transmitted his own prediction by 18 September in a letter to Johann Galle of the Berlin Observatory. The letter arrived five days later, and the planet was found with the Berlin Fraunhofer refractor that same evening, 23 September 1846, by Galle and Heinrich d'Arrest within 1° of the predicted location near the boundary between Capricorn and Aquarius.
There was, and to an extent still is, controversy over the apportionment of credit for the discovery. There is no ambiguity to the discovery claims of Le Verrier, Galle, and d'Arrest. Adams's work was begun earlier than Le Verrier's but was finished later and was unrelated to the actual discovery. Not even the briefest account of Adams's predicted orbital elements was published until more than a month after Berlin's visual confirmation. Adams made full public acknowledgement of Le Verrier's priority and credit (not forgetting to mention the role of Galle) when he gave his paper to the Royal Astronomical Society in November 1846:
I mention these dates merely to show that my results were arrived at independently, and previously to the publication of those of M. Le Verrier, and not with the intention of interfering with his just claims to the honours of the discovery; for there is no doubt that his researches were first published to the world, and led to the actual discovery of the planet by Dr. Galle, so that the facts stated above cannot detract, in the slightest degree, from the credit due to M. Le Verrier.
— Adams (1846)
Urbain Le Verrier