The OP asks 3 basic questions: First, "In what scripture(s) do the prophets say that the son of man must be put to death, and on the third day he shall rise again?" Second, "Is the scripture that Yeshua was referring to Hosea 6:1?" and Third, "Who is the Son of Man referred to in in Luke 18:31?" I will answer these questions in reverse order.
It is firstly evident in Luke 18:31, that Jesus is describing himself as the Son of Man, since it is Jesus who is delivered unto the Gentiles, spit on, scourged, put to death and rises after the second day according to Luke. But the OP is right to see that in this text, the "Son of Man" was a reference to more than just Jesus and was designed to invoke particular passages, traditions and theology. So the first question I will attempt to answer is: what is the origin of the phrase "Son of Man" and what exactly is intended to evoke to the reader?
Who or what is the "Son of Man"
According to Dr. John Nolland
As a piece of unidiomatic Greek the phrase [ὁ υἱὸς τοῦ ἀνθρώπου, “the Son of Man,”] is universally recognized to be an over-literal translation of an underlying Semitic idiom. The indefinite Hebrew form בן־אדם, ben˒ādām, “[a] son of man,” is used as God’s address to Ezekiel the prophet (Ezek 2:1; etc.; cf. Dan 8:17).
After documenting the use of our phrase to mean “a human being” and /also as an indefinite pronoun (“someone”), Vermes (“Use of בר נשׁ/בר נשׁא”) goes on to draw attention to a series of rabbinic texts in which he claims that the phrase functions as a circumlocution for “I.” He has certainly identified a use of the phrase in which the speaker intends the phrase to refer to himself, and since this appears to be what we have in the Gospel usage, such a discovery is potentially of great importance.
Jeremias (New Testament Theology, 261 n. 1) soon protested that Vermes had claimed too much. We do not have a true circumlocution for “I,” since in none of the texts does the expression have the sense “I (and no other).” A series of other scholars have raised essentially the same objection but have recognized in Vermes’ study an important foundation for further work.
Fitzmyer (“The New Testament Title,” 152–53; JSNT 4  58–64) has raised a quite different objection to Vermes’ conclusion. Fitzmyer accepts Vermes’ analysis of the idiom in the one case of Cairo Targum B to Gen 4:14, but he points to the absence of the initial א, ˒e, in every instance cited by Vermes in support of the use as a surrogate for “I.” This contrasts with Fitzmyer’s observation that in all the Aramaic evidence prior to the Second Revolt (that is before a.d. 131/2) the longer form is used. Fitzmyer warns against using what is, therefore, late evidence for identifying first-century Aramaic idiom. There is no such dating problem in the case of the use of the idiom to refer generally to “humanity (anyone)” or indefinitely to “someone” (see examples in Fitzmyer, 148).
Not everyone has been persuaded that, given the quite scant earlier Aramaic sources, we should rule out later sources for identifying likely first-century idiom. This relaxation of criteria seems more reasonable if it is not Vermes’ unique idiom that is being claimed, but rather a use of one of the more widely established idioms (general or indefinite) in a context where self-reference is involved. Casey (JSNT 29  22–23) has been able to support this relaxation by pointing to an eighth-century b.c. text (Sepire 3:14–17) in which an indefinite use of בר אנשׁ involves self-reference (“anyone of us”). Casey, Lindars (Jesus Son of Man), and Bauckham (JSNT 23  23–33) have each developed, on the basis of this sort of use of Vermes’ results, an understanding of the Aramaic idiom reflected in the Gospel usages.
Casey’s argument is that the authentic Son of Man statements are those in which Jesus makes a general statement about humanity, but in a way that invites a specific application of this general truth to himself. Lindars offers a more subtle analysis, and proposes to find a way between Vermes and Casey.
Lindars identifies an idiomatic use of the generic article as the key to Vermes’ Aramaic texts and to Jesus’ use of the Son of Man expression. To understand Lindars we need first to attend to the presence, as was mentioned above, of the suffix א, ā˒;, in some of the texts adduced by Vermes. Originally this functioned much as the definite article (“the”) in English but gradually lost its force so that forms with and without this suffix came to be used interchangeably. Vermes and most other scholars have assumed that this interchangeability was already well advanced in the time of Jesus. Lindars thinks this is not the case, even in the later rabbinic period, at least with the particular idiomatic use of the generic article with “Son of Man,” which it is his concern to establish (Fitzmyer , also, thinks that there was still, in the first century, more life than is normally allowed in the distinction between the forms with and without suffix). By generic article Lindars means the use of the suffix to “denot[e] a particular but unspecified member or group of members of the class” identified by the word or phrase to which the suffix is attached (as clarified in JSNT 23  35–36). The particular idiom that Lindars claims to have identified is the use of this generic article in speech as an indirect means for the speaker to refer to himself. The use of the suffix indicates that the speaker has a specific person in mind, and the context encourages the hearer to identify that person with the speaker himself.
Bauckham considers that Lindars has not successfully demonstrated the existence of this distinctive idiom, and opts for a much simpler suggestion. Unlike Casey, Bauckham shares Lindars’ sensitivity to the presence of the suffix as potentially affecting meaning, though, as we will see, this creates difficulties for his own position. Casey saw Jesus as achieving self-reference by making a general statement about humanity; Bauckham sees Jesus as achieving self-reference by making indefinite statements (“someone”), which he invites his hearers to recognize as pertaining to himself. Bauckham thinks that Jesus will have needed to use the form without suffix to indicate the indefiniteness. The Gospel forms are of course all definite, and this is taken by Bauckham as reflecting the translator’s wish specifically to avoid Jesus’ original ambiguity (33 n. 23).
When a general truth is being expressed, the suffix seems to be uniformly present. When an indefinite statement is being made, that is, a statement about an unspecified “someone,” then the suffix is uniformly absent. Only one precision needs to be added. As in other languages, the use of “someone” in a negative statement produces a statement of general validity: that is, the statement is true of “nobody”; or, said another way, the negation is true of any and every person.
Nolland also surveys the usage of the term noting that,
The term is used ten times in relation to a future coming of Jesus, six times in connection with the passion, seven times in reference to Jesus’ earthly life and ministry, once in reference to future glory but not to a coming (22:69), and once in a blessing on those who suffer for the sake of the Son of Man (6:22).
Luke follows Mark in establishing a certain relationship between the terms ὁ χριστός, “the Christ,” and ὁ υἱὸς τοῦ ἀνθρώπου, “the Son of Man.” While the identification of Jesus as the Christ is emphatically given in the privileged perspective of the infancy narratives (and cf. 4:41), Jesus comes forward on the story line (and also in the editorial comment 5:24) as a figure of authority and dignity under the designation “the Son of Man” (6:5, 22; 7:34). The apostolic confession of Jesus as Christ in 9:20 is treated ambivalently by Jesus (cf. at 4:41), and precipitates a quite different set of affirmations about the Son of Man (9:22, 44; 18:31–32; 22:22, 48; 24:7 and cf. 9:58 [the passion predictions]), and perhaps even a further set of affirmations (9:26; 12:8, 40; 17:22, 24, 26, 30; 18:8; 21:27, 36; cf. 22:69 [the coming of the Son of Man; for Luke, the suffering is the way to glory (24:26), which in turn is the basis for the coming (again, 19:12); the suffering Son of Man and the coming Son of Man are linked (9:22–26) on the basis that the coming Son of Man will be ashamed of those who do not confess the suffering Son of Man by following him in the way of the cross]). Meanwhile the original stream of Son of Man references continues (11:30; 12:10; 19:10). Finally, in the hour of the power of darkness (22:53), when Jesus stands as a powerless and rejected sufferer (already 22:63–64), he is questioned about being the Christ (22:67), and answers in terms of the Son of Man sitting at the right hand of God (v 68), which is understood as a claim to be the Christ (v 70 [“Son of God”; cf. 23:2 and at 3:22 and 4:41]). From this point the title “Christ” replaces “Son of Man” as the term in relation to which Jesus is described as suffering (and as being established in glory [24:26, 46; cf. Acts 3:18, etc, and the dramatic exploitation of the paradox of the suffering Christ in Luke 23:35, 39]). We are led to a suffering Christ by means of a suffering Son of Man (and probably also to a Christ of cosmic rule through a Danielic [Jewish apocalyptic?] Son of Man).
Luke’s pattern would be fitted by an understanding of Son of Man as a mysterious term of dignity with a good measure of plasticity: “The Man,” perhaps “the man, divinely raised up and given authority, with whom the destiny of humankind (Israel) is bound up.” Neither a simple periphrasis for “I” (cf. the editorial usage in 5:24), nor the exalted messianic Son of Man of apocalyptic Judaism (the shifts between Christ and Son of Man hardly fit this view, and cf. also 12:10) are adequate to the Lukan pattern; nor is Tödt’s “the traditional agent of God’s reign” (The Son of Man, 109), nor any appeal to promiscuity in the use of Christological titles (see at 4:18).
In this passage
Nolland also notes that the scene in Luke 18:31 serves a very special purpose to divide the text into three sections (which he uses to divide his commentary as well)
Brief though it is, the present unit is to be identified as constituting the terminal section for the long Journey to Jerusalem narrative, which began at 9:51. In a manner similar to what has been noted for the preceding several units, by echoing the motifs of 9:21–22, 43b–45 from the section that prepared for the journey narrative, this unit makes a contribution to the inclusio effect that surrounds the journey narrative (we end where we began). This unit is separated from the preceding by a change of audience and theme, and is separated in the same way from what is to follow.
Luke continues with his Markan source and sequence (Mark 10:32–34). He trims away the Markan setting and some of the content that would be repetitive after 9:22, and then he adds a reformulation of 9:45 (which, while based on Mark 9:32, was already largely Lukan). Other minor changes are also noted below.
It seems helpful then to view the earlier usage of the "Son of Man" in Luke 9:43b-45
But while the entire crowd was amazed at everything Jesus was doing, he said to his disciples, “Take these words to heart, for the Son of Man is going to be betrayed into the hands of men.” But they did not understand this statement; its meaning had been concealed from them, so that they could not grasp it. Yet they were afraid to ask him about this statement.
As you can see from this passage, Luke 18 is not the first time in Luke that Jesus has referred to himself as the "Son of Man" and predicted his demise. In fact, Nolland misses the earliest reference to Jesus as the "Son of Man" and places the first reference in 5:24. Instead, the first reference is actually in the genealogy of Jesus found in 3:23-38. In Hebrew, the name Adam literally means "Man" and it is interchangeable in meaning with the name "Adam".. By virtue of being the son of Adam, Jesus is to be understood by the author of Luke as the "Son of Man" and this is part of why the genealogy is provided.
Unsurprisingly, Nolland, like most westerners miss the significance and meaning of Genealogies. Anthropolgist Dr. Richard Rohrbaugh does an admirable job of elucidating this meaning to a western audience in his lecture before the Biblical Archaeology Society, explaining that Geneologies are not just important in Middle-Eastern society at that time (and now), but are the most important differentiator in society. Dr. Rohrbaugh explains that as Westerners, we live in what is known as a "guilt culture". Accordingly, we lack an understanding of the importance of familial lineage. Most of the Middle-east operates on what is know as an "Honor-Shame culture"
Now, The Baltic Culture Continent that you and I live in, anthropologist call a “Guilt Culture”. ...the Mediterranean Culture Continent both today and in antiquity is what anthropologists call an “Honor-Shame Culture”.
Honor is relatively simple actually to understand. Honor is your standing in the pecking order of the village, together with the public recognition of that. There is no such thing as claiming honor that the village does not recognize. To claim honor that the village does not recognize is to be uppity; brash; a braggart; a fool. Honor is public reputation in the village and everybody in the village knows exactly where you stand in the pecking order. The reason for that is there are two primary ways in which you can get your honor rating or ranking in a village. ...
The overwhelming way in which you get your honor rating is from your birth. It's what anthropologist call “ascribed honor.” It's the honor that you get the day you pop out of the womb. It's the honor that you and every member of your extended family has - male and female - everybody in your family has, has always had, and always will have. That kind of Ascribed Honor means that if you are born in a very high family, you have a high honor ranking. If you are born a low-life, you have a low honor ranking and you're probably going to have a till the day you die. So the overwhelming way in which you get your honor is from the family of your birth. Do you now understand why genealogies are so important in the Bible? Genealogies indicate in writing what a village knows orally. Namely, the family you were born in and hence the on a ranking you have.
I spent some time living in the little Palestinian village of Beit Jala in the West Bank doing research. My primary area of interest is peasant studies and in living in Beit Jala, I discovered that when I would be introduced to peasants they would often immediately tell me their genealogy. For most peasants it's only three generations long: me, my father, and my grandfather. Maybe my son - Maybe four. Sometimes in an oral setting like that they will go back and add the eponymous ancestors of the community - Abraham Isaac and Jacob.
Genealogies are, if you will, a kind of a map for the whole community - describing exactly where in the scheme of things you fit. Among non-literate people (which in antiquity was about ninety-six percent of the population); among non-literate people genealogies would be very short. Only upper class wealthy people have written genealogies. And you understand that the longer the genealogy, the better? Because it means you're from old money not new money. You understand?
...What I find interesting is the genealogy in Luke - It goes all the way back to “son of seth, son of Adam, son of God”. That is, it traces it to the beginning. That's the longest genealogy possible. Do you understand that in honor claim is being made? In fact we know from Roman texts that people in the Roman world who did become newly rich and wanted to move up the social ladder hired genealogists to create fictive genealogies for themselves and there were a few stars in the pantheon of Roman ancestors they all wanted to be associated with. For a fee, you could get that association. Now you have the map that tells everybody where you fit in the pecking order of things and that had an enormous impact on your life.
I’m going to show you some slides in a minute of some ancient text that describe the fact that your honor ranking determine who ate with whom, who could marry whom, who spoke to whom, who listened to whom. It determined who would speak first in a conversation. It determined who would marry whom and who would do business with whom. In fact it determined most of the social patterns of your life. It's therefore critically important that everybody in the village know exactly where you stand - because it provides the road map for how you and I are going to interact with each other.
Jesus comes from a no-account little village. He's a village – a τέκτων (tektón) he is called in the Greek. We translate it carpenter – it could be a worker in metal, stone, or wood. I don't have time today but to show you just how low on the social scale that really is - It's very near the bottom. People like that don't get up and talk in public. So when Jesus does, It confuses everybody. In the Middle-East they expect somebody born of a great family to be great. You're born of a low family you're going to be no-account. What does not compute in their social compass is somebody born to a low-life family who turns out to be great. How do you explain that?
Well of course what Matthew and Luke do to explain it is they give us these very elaborate birth stories in which they try to tell us that God was somehow unusually involved in this birth. Otherwise there could be no expectation that anybody would listen to Jesus. What do his opponents do in the twentieth chapter of Luke? They say, “Who gives you the authority to speak like this?” Note that they didn't say, “Did you have this authority from your birth?” They know that's not true. “Who gave you this authority?” Their assumption is somebody had to have acquired the honor from somebody who recognized it because all honor has to be publicly recognized.
living in Beit Jala I discovered that when I would be introduced to peasants they would often immediately tell me their genealogy. For most peasant it's only three generations long: Me, my father and my grandfather. Maybe my son - Maybe four. Sometimes in an oral setting like that they will go back and add the eponymous ancestors of the community (Abraham, Isaac and Jacob.)
Genealogies are - if you will - a kind of a map for the whole community; describing exactly where in the scheme of things you fit. Among non-literate people (which in antiquity was about ninety six percent of the population) genealogies would be very short. Only upper class wealthy people have written genealogies. And you understand that the longer the genealogy, the better - because it means you're from old money, not new money. You understand?
Anybody curious why only two of our gospels have genealogies for Jesus? Matthew and Luke provide a genealogy; Mark and John do not. What I find interesting is the one in Luke. It goes all the way back to “...son of Seth, Son of Adam, son of God.” That is, it traces it to the beginning. That's the longest. Genealogy possible. Do you understand that in honor claim is being made? In fact, we know from Roman texts that people in the Roman world who did become newly rich and wanted to move up the social ladder hired genealogists to create fictive genealogies for themselves. And there were a few stars in the pantheon of Roman ancestors that they all wanted to be associated with. For a fee, you could get that association. Now you have the map that tells everybody where you fit in in the pecking order of things and that had an enormous impact on your life.
I’m going to show you some slides in a minute of some ancient text that describe the fact that your honor ranking determine who ate with whom, who could marry whom, who spoke to whom, who listened to whom. It determine who would speak first in a conversation. It determined who would marry whom and who would do business with whom. In fact it determined most of the social patterns of your life. It's therefore critically important that everybody in the village know exactly where you stand. Because it provides the road map for how you and I are going to interact with each other.
What Old Testament passages is "Son of Man" referencing?
In reading various comments by the OP, it is apparent that he thinks that this is a single passage. However, the text states that
all things that are written by the prophets concerning the Son of man shall be accomplished. From this we can take away that there were multiple prophets who wrote multiple statements regarding the Son of Man. Instead of referencing a single passage which indicates that the Son of Man would be delivered unto the Gentiles, would be mocked, spitefully entreated, and spit upon, beaten, put him to death, and that on the third day he shall rise again, Jesus is sourcing from a multiplicity of authors and passages, each one supporting a different point from this list.
So while numerous others answers here have cited commentators stating that Jesus is referencing Hosea 6:2
After two days he will revive us; on the third day he will restore us, that we may live in his presence.
There are additional passages being referenced here as well. But before I get to those, it should be noted that Hosea seems to be using the N, N+1 Hebrew poetic standard similar to Job 5:19, (for example):
He will deliver you from six calamities; yes, in seven no evil will touch you.
As well as many other Old Testament poetic passages that use this poetic device. Just as Jesus rose after two day and was resurrected on the third day, Hosea prophesies that
After two days he will revive us; on the third day he will restore us.
In addition to this prophesy, the cross-reference for the NIV 2 indicates that those things
written by the prophets about the Son of Man include Psalm 22 which indicates the Messiah would save his people, that the Son of David would be taunted and mocked, that he would thirst, that the Son of David's enemies would cast lots for his clothes and that his hands and feet would be pinned. This passage repeatedly references the enemies of the Son of David as wild dogs, a derogatory slang term for Gentiles. The NIV also indicates that the writings of the Prophets include Isaiah 53 which indicates that the Messiah would be "despised and rejected by people" experience pain and take on the punishment for the illnesses/sins of Israel, that he would be "led away after an unjust trial" and buried in a Rich man's tomb after seeing a criminal's death.
Furthermore, Nolland connects this with the prophesy of Daniel 7, stating
A more profitable starting point is the corresponding Aramaic phrase בר אנשׁ, bar ˒enāsh. This is the phrase that occurs in Dan 7:13, which is clearly echoed in the present form of a number of the Gospel Son of Man sayings, and this is the phrase that has been the subject of a number of recent studies of Aramaic idiom that have a potential for illuminating the Gospel usage. (The phrase occurs with some variations of form, the important ones being the dropping of the א, ˒e, and/or the addition of a suffix א, ā˒.)
Ultimately, Nolland concludes that,
I start from Bauckham’s “somebody,” by noting that if the “somebody” is both somebody with whom Jesus is potentially to be identified, and at the same time a “somebody” of significance, whose precise identity remains yet unspecified, then considerably more of the texts begin to fit neatly into an approach like Bauckham’s. To modernize, we might think of something like “the Man of destiny.” This would do justice to the way in which the “Son of Man” seems regularly to be assumed to be a figure of authority and dignity, without being identified as any specific dignitary. This would also allow for the possibility of an added precision being offered at a certain point or in certain contexts: this figure is, whatever else he may also be, the man of Dan 7. Prior to that further explication, the usage would remain somewhat mysterious, and even with the further explication we are not to understand that the contours of the phrase are exhaustively given in the Dan 7 imagery or the traditions associated with it in its contemporary usage. At times there could be real doubt as to the application to Jesus himself. If he were to be taken as referring to himself, then the claim of the idiom would be no more precise than to be a figure of importance. But all the time the idiom would be open to the precision to come with the eschatological role identified in the link to Dan 7, but not in the sense that “Son of Man” is now recognized to have been all along a cryptic reference to Dan 7:13. As is not the case with Bauckham, who needed to find an underlying form without the suffix, the usage suggested here would be better expressed with the form that uses the suffix. This is not exactly any existing Aramaic idiom. It would be a distinct coinage remembered by the early church as a characteristic diction of Jesus, in much the way that the anticipatory amen was remembered or the use of Abba.
The Talmud represents the oldest written records of rabbinic tradition. It contains the Mishna which Wikipedia characterizes as "a written compendium of Rabbinic Judaism's Oral Torah." As the Mishna is written in Aramaic, just like the book Daniel, it is very important that the Talmud identifies the "Son of Man" as being the Messiah. While we do not know for sure that the idea of the Messiah predates Jesus, the odds are pretty good that it does, and this is accepted by both Judaism and Christianity (though they disagree on who that Messiah was). Similarly, the Talmud and rabbinic tradition accept Psalms 22, Isaiah 53 and Hosea 6:2 as references to the Messiah.
While no particular passage says that "the son of man must be put to death" and on the third day he shall rise again, some scriptures do tell us that 1) the messiah would be put to death and 2) others tell us that that same messiah would be raised up on the 3rd day.
We see that indeed, Luke is referencing Hosea 6:2, but he is also referencing Psalm 22, Isaiah 53 and Daniel 7.
It is clear, from this text that Jesus is claiming to be the Son of Man and that Luke expects his readers to understand based on these passages and rabbinic tradition that the Son of Man is the Messiah; Christ.
1 Nolland, John: Word Biblical Commentary : Luke 9:21-18:34. Dallas : Word, Incorporated, 2002 (Word Biblical Commentary 35B), S. 469-473
2 Be sure to click the "Page Options" gear and check "Cross-Reference"