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Second favourite after Aso, whom he succeeded as Foreign Minister in August. Machimura, 62, is head of the largest faction inside the LDP.
Vice chairman of a parliamentary group promoting friendly relations with China, Machimura criticised former Premier Junichiro Koizumi’s visits to the Yasukuni shrine, which honours both war dead and war criminals and became a symbol of tensions between Japan and its neighbours.
He too did post-graduate studies in the United States, and is from a political family - he took over his father's constituency on the island of Hokkaido in 1983.
Solidly experienced in government, he was made Education Minister in 1997 and after that led Japan's Science and Technology Agency.
Lloyd-Parry's verdict: "A possible, I would probably place him second after Aso."
Sadakazu Tanigaki
A former Cabinet Minister and now head of a minor LDP faction, Tanigaki too has stood unsuccessfully for the premiership before, and is also the son of a parliamentarian. Seen as uninspiring and lacking the charisma of his rivals.
The 62-year-old served as Deputy Finance Minister, headed the national commission for reviving the financial industry and held other portfolios before becoming Finance Minister under Koizumi, vowing to reform Japan's troubled state finances.
Has publicly vowed not to visit the Yasukuni shrine, and advocates a moderate foreign policy to improve relations with China and South Korea.
Lloyd-Parry's verdict: "I would be surprised if he made it unless the party decided it just wanted to go for someone safe and boring."
Fukushiro Nukaga
Appointed Finance Minister only last month, Nukaga, 63, is a former journalist for a conservative newspaper. Has twice headed the Defence Ministry, but on the first occasion in 1998 he lasted only three months, before stepping down amid a corruption scandal involving defence procurement.
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