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According to Ganesha Purana, there are four incarnations of Lord Ganesha:
According to Mudgal Purana, there are eight avatars of Lord Ganesha:
Vakratunda, Ekadanta, Mahodara, Gajanana, Lambhodara, Vikata, Vighnaraja, Dhoomravarna.
Now I am totally confused as to which Purana is right.
I don't think there is any problem if both the list are simultaneously correct. :)
For the time being I can give you the relevant verses from the Mudgala Purana. According to this Purana Lord Ganesha took 8 incarnations to kill 8 specific Daityas.
Quoting from the article "Ganesha Mahima":
Mudgala Purana details various manifestations of Ganesha as of killing Daityas:
Vakratundaavataarascha dehaanaam Brahma thaarakah,
Matsaraasura hantaa cha simhavaahanagah smtitah/
Ekadantaavataaro vai dehaanaam Brahmadhaarakah,
Madaasurasya hantaa sa aakhuvaaha -nagah smritah/
Mahodara iti khyaato Jnaana Brahma prakaashakah,
Mohaasurasya shatruvaim aakhu- vaahanagah smritah/
Gajaananah sa vigjneyah saankhebhyah siddhi daayakah,
Lobhaasura prahartaa vai aakhugascha prakeertitah/
Lambovataaro vai Krodhaasura nibarhanah,
Shakti Brahmaakhugah sad yat tasya dhaaraka uchyate/
Vikato naama vikhyaatah Kaamaasura vidaahakah,
Mayura vaahanash -chaayam Sourabrahmadharah smritah/
Vighnaraajaavataarascha shesha vaahana uchyate,
Mamata--asura hantaa cha Vishnubrahmoti vaachakah/
Dhumra varnaavataarascha abhimaanaasura naashakah,
akhuvaahana evaasou Shivaatmaatu sa uchyate ||Vakratunda sporting the back on a roaring lion killed Matsaraasura. Seated comfortalby as a ‘mooshaka vaahana’, the Lord with His ‘ekadanta’ the single tusk destroyed Madaasura. Mooshaka vahana Ganesha as manifested as Mahodara killed Mohaasura.Assuming the title of Gajaanana killed Lobhaasura while as Lambodara destroyed Krodhaasura. The outstanding evil of the Society viz. Kaamaasura was subdued and killed by Ganesha as Mayura Vahana while as Sesha Vahana Vighna Raja killed Mamataasura. As Mooshaka Vaahana, Maha Ganesha destroyed Abhimaanaasura. Thus as Vakratunda-Ekadanta and such ‘avataaraas’, Ganesha destroyed Matsara, Mada, Moha, Lobha, Krodha, Kaama, Mamata and Ahankaara naamaka asuras.
So as Vakratunda he killed Matsarasura, as Ekadanta Madasura etc.
(Note - The author of the cited article is the same person who's books get published on the Kamakoti.org. All the "Essence of Purana Series" books are authored by him.)
