Gemara

6b - Question: From when do we count the year of a Bechor (in which it must be offered or eaten)?

Answer #1 (Abaye): We count from its birth.

Answer #2 (Rav Acha bar Yakov): We count from when it may be offered, i.e. from the eighth day.

They do not argue. Rav Acha bar Yakov discusses a Tam, and Abaye discusses a Ba'al Mum.

Question: Why do we count a Ba'al Mum from its birth? One cannot eat it (before eight days, because it is Safek Nefel)!

Answer: The case is, Kalu Chadashav (we know that it (is not a Nefel, for it) had a full term pregnancy).

5b - Beraisa - Others - Suggestion: A Bechor that was not offered in its first year should be Pasul like a blemished Korban!

Rejection: "V'Ochalta... Ma'asar Degancha... u'Vchoros Bekarcha" equates Bechor and Ma'aser (of grain). Just like Ma'aser is not Nifsal if it is not eaten within the year, also Bechor.

Temurah 21b: Other Tana'im learn this from "Lifnei Hash-m Elokecha Sochalenu Shanah v'Shanah."

Bechoros 26b - Mishnah: A Bechor should be eaten within its (first) year, whether it is Tam or a Ba'al Mum - "Lifnei Hash-m Elokecha Sochalenu Shanah v'Shanah."

If it developed a Mum within its year, he may keep it all 12 months. After its year, he may keep it for only 30 days.

28a: Question: How do we understand the Seifa?

Perhaps it is all one clause. If the Mum came within its year, he may keep it for 12 months and 30 days (from its birth);

Or, perhaps is it two clauses. If the Mum came within its year, he may keep it until 12 months (Rashi - from its birth; Shitah - from when the Mum came). If the Mum came after its year, he may keep it for 30 days?

Answer #1 (Beraisa): Nowadays, before one has the opportunity to show a Bechor to a Chacham, one may keep it for two or three years;

After he has the opportunity to show it to a Chacham, If the Mum came within its year, he may keep it until 12 months; after its year, he should be required to slaughter it immediately. However, to avoid a loss, he may keep it for 30 days.

Rejection: Also the Beraisa can be understood either way!

(Perhaps it is two clauses - if the Mum came within its year, he may keep it until 12 months; if the Mum came after its year, he may keep it for 30 days (to avoid a loss);

Or, (it is all one clause;) the Mum came within its year - he may keep it 30 extra days after 12 months.

Answer (Beraisa): If a Mum came 15 days before the end of its year, he may keep it until 15 days after the year, to complete 30 days.

This shows that the Mishnah is two clauses. Thirty days are allowed for a Mum that comes after the year.

Rishonim

Rambam (Hilchos Bechoros 1:8): A Bechor must be eaten within its year, whether it is Tam or Ba'al Mum - "Lifnei Hash-m Elokecha Sochalenu Shanah v'Shanah; V'Chi Yihyeh Vo Mum... bi'Sh'arecha Tochlenu". If it is Tam we count the year from day eight, when it was fitting to be offered. If it was born with a Mum, and Kalu Chadashav, we count the year from its birth, for it was permitted immediately. If we do not know that Kalu Chadashav, we count the year from day eight.

Me'iri (6b DH ha'Bechor): If a Bechor was born with a Mum we count its year from birth. This is even if we could not eat it before day eight lest it is a Nefel, because in some cases it is permitted immediately. The other Poskim disagree.

Tosfos (Bechoros Sof 20b): Mid'Oraisa, a baby within 30 days is not a Safek Nefel, for the majority of babies are viable. Mid'Rabanan we are stringent like R. Shimon ben Gamliel, and call it a Safek. We are similarly stringent about an animal within eight days, for otherwise people would not follow the ruling about people.

Turei Even (Rosh Hashanah 7a): Mid'Oraisa, "Sochalenu Shanah v'Shanah" applies only to a Tam, which is eaten "Lifnei Hash-m". We count the year of a Ba'al Mum from when it is permitted, i.e. from eight days unless Kalu Chodoshav. Mid'Oraisa it is always permitted immediately (like Tosfos)! How can an Isur mid'Rabanan until eight days extend the year to eat it? We must say that the Chiyuv to eat it within a year is only mid'Rabanan.

Kuntres Acharon (to Turei Even ibid.): The Yerushalmi explicitly says that the Chiyuv to eat it within a year is only mid'Rabanan.

Rambam (12): The year for a Bechor is a full lunar year of 12 months from day to day. If it was a leap year, it is 13 months.

The Rosh (Bechoros 4:2) brings the Mishnah and the conclusion of the Gemara there and in Rosh Hashanah.

Rosh (ibid.): If one transgressed and kept a Bechor after its year it is not Pasul.

Poskim

Shulchan Aruch (YD 306:7): A Bechor, even a Ba'al Mum, must be eaten within its year.

Taz (6): This is so one will not transgress shearing it or working with it (these are forbidden even if it has a Mum).

R. Akiva Eiger (DH b'Gizah): This is like the Yere'im and Levush, who say that the verse ("Shanah v'Shanah") applies to a Tam, but the Isur regarding a Ba'al Mum is mid'Rabanan. The Rambam and SMaG explicitly says that it is mid'Oraisa even regarding a Ba'al Mum.

Rashash (Bechoros 28a DH Aval): Teshuvas Shivas Tziyon (p24) learns from Bechoros 28a that the Mitzvah to eat a Ba'al Mum within the year is mid'Rabanan. The Gemara said that we always allow 30 days from when the Mum came, even if this extends past the year, because of Hashavas Aveidah. This is not a proof. The Rambam holds that mid'Oraisa one has a year from when the Mum came, just like one has a year for a Tam from when it could be offered. Mid'Rabanan, one has only until a year from birth. Because of Hashavas Aveidah, one always gets at least 30 days.

If it was born with a Mum, and Kalu Chadashav, we count the year from its birth, for it was permitted immediately. If we do not know how long its gestation, we count the year from day eight (the first day one may eat it).

Shulchan Aruch (8): If the Mum came during the year, one may keep it all 12 months.

Shach (14): The Tur says that when the Mum came later, one may keep it 12 months from its birth. The Beis Yosef explains that this is if Kalu Chadashav. Otherwise, we count the year from day eight. This is unlike Rashi and Tosfos, who explicitly say that we count from birth only if Kalu Chadashav and it was born with a Mum. It seems that the Rambam and SMaG agree. The Gemara supports them, for it asked why we count a Ba'al Mum's year from its birth, before it may be eaten. To answer this we must say that Kalu Chadashav and it was born with a Mum! It seems that our text of the Tur is mistaken. It should say 'one may keep it 12 months from the eighth day from its birth.' The previous words of the Tur prove that he holds like this; Kalu Chadashav does not help, since it was born Tam and could not be eaten immediately anyway.