The answer is there is a very weak (fabricated) narration that creates a relation between fasting these days and 'Adam and Hawaa'.
Imam as-Suyuti referred to it in his tafsir a-Durr al-Manthur الدر المنثور -see here- while commenting (2:37) saying:
In the following I'm translating text from Arabic language, as this translation is of my own take it carefully! .
وأخرج الخطيب في أماليه وابن عساكر بسند فيه مجاهيل عن ابن مسعود عن النبي صلى الله عليه وسلم قال إن آدم لما أكل من الشجرة أوحى الله إليه: اهبط من جواري
And al-Khatib (al-Baghdadi) reported in al-Amaaly (bi Jami' Dimashq) and ibn 'Asakir with a chain of narrators in which there are unknowns on the authority of ibn Mas'ud from the Prophet, may Allah’s prayers and blessings be upon him, who said that when Adam ate from the tree, Allah revealed to him: Go down from my neighborhood.
وعزتي لا يجاورني من عصاني فهبط إلى الأرض مسودا فبكت الأرض وضجت
فأوحى الله: يا آدم صم لي اليوم يوم ثلاثة عشر
By My pride none who disobeys Me stays adjacent to Me,
So he fell to the ground dark faced, and the earth cried and raged,
Allah then revealed: Oh Adam, fast for Me today on the thirteenth day
فصامه فأصبح ثلثه أبيض ثم أوحى الله إليه: صم لي هذا اليوم يوم أربعة عشر
فصامه فأصبح ثلثاه أبيض ثم أوحى الله إليه صم لي هذا اليوم يوم خمسة عشر
فصامه فأصبح كله أبيض
فسميت أيام البيض
He fasted, and a third of it was white, then God revealed to him: Fast me this day on the day of fourteen.
He fasted, and two thirds of it became white, then God revealed to him to fast for me this day on the fifteenth day.
He fasted, and it was all white.
It was called the days of eggs
Ibn al-Jawzi mentioned this hadith in his al-Mawdu'aat (the fabrications) كتاب الموضوعات -see here- and commented it saying:
هذا حديث لا يشك في وضعه وفي إسناده جماعة مجهولون لا يعرفون.
وإنما سميت أيام البيض لأن الليل كله يبيض بالقمر
This is a hadith whose status and its chain of transmission are not a subject of doubt of it being fabricated, as in its chain is a group of unknown persons who are not known.
In fact it is called the white days because the whole night is whitened by the moon.
Therefore this narration must be regarded as a fabrication.