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Leo Szilard invented the atomic fission bomb in 1933 in London while crossing a street. He applied for a patent on it in 1934 and was granted that patent (GB630726) in 1936, at which time the British Admiralty classified the patent to prevent Nazi Germany from seeing it, the patent remained classified until 1949. Szilard worked on the Manhattan Project at Los Alamos helping develop the atomic fission bomb.

It is unclear exactly who invented the hydrogen fusion bomb as it came up in many group discussions early in the Manhattan Project. Edward Teller liked to have people believe he was the inventor, but that is unlikely. However Teller became fixated on it (calling it the Super Bomb) to the point that he began to hinder other people's work on the atomic fission bomb and finally Oppenheimer had to limit Teller's access to others on the project (which Teller never forgave him for). Before the war ended Teller had completed a hydrogen fusion bomb design that he called the Classical Super but was not allowed to try building it. However in December 1945 and January 1946 a simulation of Teller's Classical Super design was run on the ENIAC, but it showed the design was an unworkable failure and serious work on hydrogen fusion bombs effectively died at that time (although Teller persisted at working alone on them in his spare time).

In 1949 a team working on atomic fission bomb improvements at Los Alamos developed an idea to improve the core compression of an atomic fission bomb (and thus its efficiency and yield) using a concept called "staging", where a small atomic fission bomb would be used to compress a second larger atomic fission bomb thus giving a much higher yield. However the team ran into problems developing the equations for a computational model of this design. They sent their mathematician Stanislaw Ulam to consult with Edward Teller on these equations. Teller immediately saw that this "staging" concept was the feature needed to get the high compression and temperature needed to finally make his hydrogen fusion bomb design workable. By 1950 working together Stanislaw Ulam & Edward Teller had invented the modern staged hydrogen fusion bomb design (usually called the Teller-Ulam configuration, although Teller kept trying to denigrate and diminish Ulam's contributions).

Note: H. G. Wells created the name "atomic bomb" for his 1914 Science Fiction novel The World Set Free, but did not have anything to do with their invention.

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Q: Who invented the atomic bomb and the hydrogen bomb?
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