David Cameron is to appear on US talk show Letterman later this week, but his biggest concern will not be whether he can handle the questions of one of the most acerbic interviewers in US TV, but whether he can outdo his great political rival Boris Johnson.

By appearing on the show, the PM is hoping to raise the profile of the UK post-Olympics, as well as push for more foreign aid to help reduce global poverty. However, Letterman could be a risky move for Cameron, with the razor-sharp host unafraid to ask uncomfortable questions.

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Only last month, London mayor Johnson was invited onto the illustrious show, which boasts more than 3m viewers each episode, to chat about London, Britain and the Olympics. Boris not only survived but talked warmly about the Jubilee and his New York heritage, charming the host, his audience and the public at large. Johnson even enjoyed some gentle ribbing from the veteran comedian for “cutting his own hair”. Cameron will be hoping for the same.

This week's appearance isn't the first time the prime minister has tread so quickly in the unkempt footsteps of the mayor, with the PM's appearance likely to fuel the rivalry between the men. Boris is, after all, a popular if unlikely alternative to Cameron's Tory leadership.

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It’s a competition that stretches all the way back to the wood-paneled private dining rooms of Oxford and the Bullingdon Club, yet the rivalry has intensified in recent years, with BoJo seemingly ahead of his the PM at every turn.

During the Olympics, Johnson delivered a stirring speech to the athletes at the start of the Games, and rebuffed Mitt Romney’s wayward comments on London’s preparedness for the Games. Both left Cameron looking flat-footed.

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The mayor, like his predecessor ken Livingstone, has also regularly used the Tube to get around. Cameron has done likewise, while the sporting rivalry between the pair was obvious during a public tennis match in the build up to the Olympics.

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Since taking office, Johnson has became well known for cycling to work, his love of the pedal emphasised by the introduction of the popular Boris Bikes in the capital. Cameron too has joined in saddle-bound revolution, being frequently photographed on his two-wheeler in and around London.

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In the coming months, the political match-up is likely to renew afresh with Johnson, a firm favourite with grass roots Tories, expected to be the star attraction at the Tory Party conference next month… much to the prime minister’s chagrin.

On the international stage, both Cameron and Johnson have trips planned to Brazil, ahead of the 2014 World Cup and the 2016 Olympics Games in Rio.

Whether Cameron, hamstrung slightly by weighty matters of state, can be as gregarious as Johnson on a chat show will be an interesting test.

Should he fail, political commenters, alongside the Tory rank and file, will be only too quick to point it out.

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